1. (a) State factor theorem.[br]Solution:[br]If [math] x-a [/math] is a factor of [math] f(x) [/math] then [math] f(a) = 0 [/math][br]OR[br]If [math] f(a) = 0 [/math] then [math] x-a [/math] is a factor of [math] f(x) [/math].[br][br]1. (b) If [math] (x-a) [/math] is a factor of [math] x^n - a^n [/math], what is the degree of quotient.[br]Solution:[br]The degree of quotient is [math] n - 1 [/math].[br][br]2. In each of the following, use factor theorem to find whether [math] g(x) [/math] is a factor of polynomial [math] f(x) [/math] or not?[br](a) [math] f(x) = x^3 + 9x^2 + 27x + 27; \ \ \ g(x) = x + 3 [/math][br]Solution:[br]Here, [math] f(x) = x^3+9x^2+27x+27 [/math][br]And [math] g(x) = x+ 3 [/math][br]Zero of [math] x+3 [/math] is [math] - 3 [/math][br][math] \begin{align} \text{ Remainder } & = f (-3) \\ & = (-3)^3+9(-3)^2+27(-3) + 27 \\ & = -27 + 81 - 81 + 27 \\ & = 0 \end {align} [/math] [br][math] \therefore \ g(x) [/math] is a factor of polynomial [math] f(x).[/math][br][br](b)[math] f(x) = x^3 + x^2 + 27x + 27, g(x) = x + 3 [/math][br]Solution:[br]Here, [math] f(x) = x^3+x^2+27x+27 [/math][br]And [math] g(x) = x+3 [/math] [br]Zero of [math] x+3 [/math].[br][math] \begin{align} \text{Remainder } & = f(-3) \\ & = (-3)^3+(-3)^2+27(-3)+27\\ & = -27 +9 - 81 +27 \\ & = -72 \neq 0 \end{align} [/math][br][math] \therefore g(x) [/math] is not a factor of polynomial [math] f(x) [/math]. [br][br](c) [math] f(x) = x^3 + 6x^2 + 7x + 9, g(x) = x – 2 [/math][br]Solution:[br]Here, [math] f(x) = x^3+6x^2+7x+9 [/math][br]And [math] g(x) = x-2 [/math][br]Zero of [math] x-2 [/math] is 2.[br][math] \begin{align} \text{Remainder } & = f(2) \\ & = 2^3+6(2)^2+7(2)+9 \\ & = 8 + 24 + 14 + 9 \\ & = 55 \neq 0 \end{align} [/math][br][math] \therefore g(x) [/math] is not a factor of polynomial [math] f(x) [/math]. [br][br](d) [math] f(x) = 3x^3 + x^2 – 20x + 12, g(x) = 3x – 2 [/math][br]Solution:[br]Here, [math] f(x) = 3x^3 + x^2 – 20x + 12 [/math][br]And [math] g(x) = 3x – 2 [/math][br]Zero of [math] 3x – 2 [/math] is [math] \frac{2}{3} [/math].[br][math] \begin{align} \text{Remainder } & = f\left(\frac{2}{3}\right) \\ & = 3\left(\frac{2}{3}\right) ^3+\left(\frac{2}{3}\right) ^2-20\left(\frac{2}{3}\right) +12 \\ & = 3\left(\frac{8}{27}\right) + \frac{4}{9}-\frac{40}{3} +12 \\ & = \frac{8+4-40\times 3 + 12\times 9 }{9} \\ & = \frac{8+4-120+180}{9} \\ & = \frac{0}{9} \\ & = 0 \end{align} [/math][br][math] \therefore g(x) [/math] is a factor of polynomial [math] f(x) [/math]. [br][br](e) [math] f(x) = 8x^3 – 4x^2 + 7x + 9; g(x) = 2x + 1 [/math][br]Solution:[br]Here, [math] f(x) = 8x^3 – 4x^2 + 7x + 9 [/math][br]And [math] g(x) = 2x + 1 [/math][br]Zero of [math] 2x + 1 [/math] is [math] \frac{-1}{2} [/math].[br][math] \begin{align} \text{Remainder } & = f\left(\frac{-1}{2}\right) \\ & = 8\left(\frac{-1}{2}\right)^3-4\left(\frac{-1}{2}\right)^2+7\left(\frac{-1}{2}\right)+9 \\ & = 8\left(\frac{-1}{8}\right) -4\left(\frac{1}{4}\right)-\frac{7}{2}+9 \\ & = -1-1-\frac{7}{2}+9 \\ & = \frac{-2-2-7+18}{2} \\ & = \frac{7}{2} \neq 0 \end{align} [/math][br][math] \therefore g(x) [/math] is not a factor of polynomial [math] f(x) [/math]. [br][br]3. (a) Find the value of [math] k [/math] , if [math] x + 3 [/math] is a factor of [math] 3x^2 + kx + 6 [/math][br]Solution:[br]Let, [math] f(x) = 3x^2+kx+6 [/math] [br]Zero of [math] x+3 [/math] is -3.[br]As [math] x+3 [/math] is a factor of f(x),[br][math] \begin{align} & \text{Remainder } = 0 \\ & \text{ or, } f(-3 ) = 0 \\ & \text{ or, } 3(-3)^2+k(-3)+6=0 \\ & \text{ or, } 27-3k+6=0 \\ & \text{ or, } 33=3k \\ & \text{ or, } k=\frac{33}{11} \\ & \therefore k=11 \end{align} [/math] [br][br][br]3. (b) Find the value of [math] k [/math], if [math] x + 1[/math] is a factor of [math] x^3 – kx^2 – 3x – 6 [/math] [br]Solution:[br]Let, [math] f(x) = x^3 – kx^2 – 3x – 6 [/math] [br]Zero of [math] x+1 [/math] is -1.[br]As [math] x+3 [/math] is a factor of f(x),[br][math] \begin{align} & \text{Remainder } = 0 \\ & \text{ or, } f(-1 ) = 0 \\ & \text{ or, } (-1)^3-k(-1)^2-3(-1)-6 =0 \\ & \text{ or, } -1-k+3-6=0 \\ & \text{ or, } -k-4=0 \\ & \text{ or, } -k=4 \\ & \therefore k=-4 \end{align} [/math] [br][br]3. (c) Find the value of [math] m [/math], for which [math] 2x^4 – 4x^3 + mx^2 + 2x + 1 [/math] is exactly divisible by [math]1 – 2x.[/math][br]Solution:[br]Let, [math] f(x) = 2x^4 – 4x^3 + mx^2 + 2x + 1 [/math] [br]Zero of [math] 1 – 2x [/math] is [math] \frac{1}{2} [/math] .[br]As [math] 1 – 2x [/math] is a factor of f(x),[br][math] \begin{align} & \text{Remainder } = 0 \\ & \text{ or, } f( \frac{1}{2}) = 0 \\ & \text{ or, } 2\left( \frac{1}{2} \right)^4 -4 \left( \frac{1}{2} \right)^3 +m\left( \frac{1}{2} \right)^2 +2\left( \frac{1}{2}\right)+1=0 \\ & \text{ or, } 2\times \frac{1}{16} -4\times \frac{1}{8} +m\times \frac{1}{4}+1+1=0\\ & \text{ or, } \frac{1}{8}-\frac{1}{2}+\frac{m}{4}+2=0 \\ & \text{ or, } \frac{1-4+2m+16}{8} = 0 \\ & \text{ or, } [br] 2m+13 = 0 \\ & \text{ or, } 2m= -13 \\ & \therefore m = - \frac{13}{2} \end{align} [/math] [br][br]4. Factorize the following by using factor theorem.[br](a) [math] 2x^3 + 3x^2 – 3x – 2 [/math] [br]Solution:[br]Let, [math] f(x) = 2x^3 + 3x^2 – 3x – 2 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2 [/math][br][math] \begin{align} \text{Now}, \\ f(1) & = 2(1)^3+3(1)^2 -3(1)-2 \\ & = 2 +3-3 -2 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = 2x^2+5x+2 \\ & = 2x^2 +(4+1)x +2 \\ & = 2x^2 +4x+x+2 \\ & = 2x(x+2) +1 (x+2) \\ & = (x+2)(2x+1) \\ \therefore f(x) & = (x-1) (x+2) (2x+1) \end{align} [/math][br][br]4(b) [math] x^3 + 2x^2 - x -2 [/math][br]Solution:[br]Let, [math] f(x) = x^3 + 2x^2 - x -2 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2 [/math][br][math] \begin{align} \text{Now}, \\ f(1) & = (1)^3+2(1)^2 -1-2 \\ & = 1+2-1-2 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2+3x+2 \\ & = x^2 +(2+1)x +2 \\ & = x^2 + 2x +x +2 \\ & = x(x+2)+1(x+2) \\ & = (x+2)(x+1) \\ \therefore f(x) & = (x-1)(x+2)(x+1) \end{align} [/math][br][br]4(c) [math] y^3 - 6y^2 + 3y + 10 [/math][br]Solution:[br]Let, [math] f(y) = y^3 - 6y^2 + 3y + 10 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2 , \pm 5, \pm 10 [/math][br][math] \begin{align} \text{Now}, \\ f(2) & = 2^3-6(2)^2+3(2)+10 \\ & = 8-24+6+10 \\ & = 0 \end{align} [/math][br][math] \therefore y -2 [/math] is a factor of [math] f(y) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math]\begin{align} \text{Quotient } &= y^2-4y-5\\ &= y^2-(5-1)y-5\\ &= y^2-5y+y-5 \\ &= y(y-5)+1(y-5)\\ &= (y-5)(y+1)\\ \therefore f(y) &=(y-2)(y-5)(y+1) \end{align}[/math][br][br]4(d) [math] x^3 + 13x^2 + 32x + 20 [/math] [br]Solution:[br]Let, [math] f(x) = x^3 + 13x^2 + 32x + 20 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 [/math][br][math] \begin{align} \text{Now}, \\ f(-2) & = (-2)^3+13(-2)^2+32(-2)+20 \\ & = -8+52-64+20 \\ & = 0 \end{align} [/math][br][math] \therefore x+2 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2 + 11x+ 10 \\ & = x^2 + 10x + x + 10 \\ & = x(x+10) + 1( x+10) \\ & = (x+10)(x+1) \\ \therefore f(x) & = (x+1)(x+2)(x+10) \end{align} [/math][br][br]4(e) [math] 2x^3 + x^2 – 2x – 1 [/math][br]Solution:[br]Let, [math] f(x) = 2x^3 + x^2 – 2x – 1 [/math][br]Possible factors of 2 are [math] \pm 1, \pm \frac{1}{2} [/math][br][math] \begin{align} \text{Now}, \\ f(1) & = 2(1)^3+1^2-2(1)-2 \\ & = 2+1-2-1 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient} & = 2x^2+3x+1 \\ & = 2x^2 + 2x + x + 1 \\ & = 2x(x+1) + 1(x+1) \\ & = (x+1)(2x+1) \\ \therefore f(x) & = (x-1)(x+1)(2x+1) \end{align} [/math][br][br]4(f) [math] x^3 – 23x^2 + 142x – 120 [/math][br]Solution:[br]Let, [math] f(x) = x^3 – 23x^2 + 142x – 120 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2, \pm 3, \pm 5 \text{ etc } [/math][br][math] \begin{align} \text{Now}, \\ f(1) & = 1^3-23(1)^2+142(1)-120 \\ & =1 -23 + 142 -120 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2 - 22x + 120 \\ & = x^2 - 12x - 10x +120 \\ & = x(x-12) -10(x-12) \\ & = (x-12)(x-10) \\ \therefore f(x) & = (x-1)(x-10)(x-12) \end{align} [/math][br][br]4(g) [math] (x – 1) (2x^2 + 15x + 15) – 21 [/math] [br]Solution:[br][math] \begin{align} \text{Let, } f(x) & = (x – 1) (2x^2 + 15x + 15) – 21\\ & = 2x^3 +15x^2+15x-2x^2-15x-15-2 \\ & = 2x^3+13x^2-36 \end{align} [/math][br]Possible factors of 36 are [math] \pm 1, \pm 2, \pm 3, \pm 4 \text{ etc } [/math][br][math] \begin{align} \text{Now}, \\ f(-2) & = 2(-2)^3+13(-2)^2-36 \\ & =-16+52-36 \\ & = 0 \end{align} [/math][br][math] \therefore x+2 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = 2x^2+9x-18 \\ & = 2x^2 +12x - 3x - 18 \\ & = 2x(x+6) -3 (x+6) \\ & = (x+6)(2x-3) \\ \therefore f(x) & = (x+2)(x+6)(2x-3) \end{align} [/math][br][br]5. Use factor theorem and solve for [math] x [/math].[br](a) [math] x^3 – 4x^2 – 7x + 10 = 0 [/math] [br]Solution:[br]Let [math] f(x) = x^3-4x^2-7x+10 [/math][br]Possible factors of 10 are [math] \pm 1, \pm 2, \pm 5, \pm 10 [/math][br][math] \begin{align} \text{Now, } f(1) & = 1^3-4(1)^2-7(1)+10 \\ & = 1-4-7+10 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2 - 3x -10 \\ & = x^2 -5x + 2x - 10 \\ & = x(x-5) + 2(x-5) \\ & = (x-5)(x+2) \\ \therefore f(x) & = (x-1)(x-5)(x+2) \\ \text{or, } 0 & = (x-1)(x-5)(x+2) \end{align} [/math][br][math][br]\begin{tabular}{|l |l |l | }[br]\hline[br]\text{Either} & \text{Or} & \text{Or} \\[br]x-1=0 & x-5 = 0 & x+2 =0 \\[br]\text{or, } x =1 & \text{or, } x=5 & \text{or, } x= -2 \\[br]\hline[br]\end{tabular}[br][/math][br][math] \therefore x= 1, 5, -2 [/math][br][br]5(b) [math] x^3 + 4x^2 + x – 6 = 0 [/math][br]Solution:[br]Let [math] f(x) = x^3+4x^2+x-6 [/math] [br]Possible factors of 6 are [math] \pm 1, \pm 2, \pm 3, \pm 6 [/math][br][math] \begin{align} \text{ Now, } f(1) & = 1^3+4(1)^2+1-6 \\ & = 1+4+1 -6 \\ & = 6-6 \\ & = 0 \end{align} [/math] [br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{ Quotient } & = x^2 +5x+6 \\ & = x^2 +3x+2x+6 \\ & = x(x+3) +2 (x+3) \\ & = (x+3)(x+2) \\ \therefore f(x) & = (x-1)(x+2) (x+3) \\ \text{or, } 0 & = (x-1)(x+2)(x+3) \end{align} [/math][br][math] \begin{tabular} { | l | l | l | } \hline \text{ Either } & \text{Or } & \text { Or } \\ x-1 =0 & x+2= 0 & x+3 = 0 \\ \text {or, } x=1 & \text{or, } x=-2 & \text {or, } x = -3 \\ \hline \end{tabular} [/math][br][math] \therefore x = 1, -2, -3 [/math][br][br]5(c) [math] 3x^3 – x^2 – 3x + 1 = 0 [/math][br]Solution:[br]Let [math] f(x) = 3x^3-x^2-3x+1 [/math][br]Possible factors are [math] \pm 1, \pm \frac{1}{3} [/math][br][math] \begin{align} \text{Now, } f(1) &= 3(1)^3-1^2-3(1)+1 \\ & = 3-1-3+1 \\ & = 0 \end{align} [/math][br][math] \therefore x - 1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient} & = 3x^2 +2x-1 \\ & = 3x^2+3x-x-1 \\ & = 3x(x+1) -1(x+1) \\ & = (x+1)(3x-1)\\ \therefore f(x) & = (x-1)(x+1)(3x-1) \\ \text{or, } 0 & = (x-1)(x+1)(3x-1) \end{align} [/math][br][math] \begin{tabular}{ | l | l | l | } \hline \text{Either} & \text{ Or} & \text{Or } \\ x-1 =0 & x+1 = 0 & 3x-1 = 0 \\ \text{or, } x=1 & \text{or, } x=-1 & \text{or, } x =\frac{1}{3}\\ \hline \end{tabular} [/math][br][math] \therefore x= \pm 1, \frac{1}{3} [/math] [br]5(d) [math] x^3-3x^2-9x-5=0 [/math] [br]Solution:[br]Let, [math] f(x) = x^3-3x^2-9x-5=0 [/math][br]Possible factors of 5 are [math] \pm 1, \pm 5 [/math][br][math] \begin{align} f(-1) & = (-1)^3-3(-1)^2-9(-1)-5 \\ & = -1-3+9-5 \\ = & 0 \end{align} [/math] [br][math] \therefore x +1 [/math] is a factor of [math] f(x) [/math][br]Now using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2-4x-5 \\ & = x^2 -5x+x-5 \\ & = x(x-5) + 1(x-5) \\ & = (x-5)(x+1) \\ \therefore f(x) & = (x+1)(x+1)(x-5) \\ \text{or, } 0 & = (x+1)(x+1)(x-5) \end{align} [/math][br][math] \begin{tabular} {|l|l|l|} \hline \text{Either} & \text{Or } & \text{Or } \\ x+1=0 & x+1 =0 & x-5= 0 \\ \text{or, } x= -1 & \text{or, } x= -1 & \text{or, } x= 5 \\ \hline \end{tabular} [/math][br] [br]5(e) [math] x^3 – 3x^2 – 10x + 24 = 0 [/math][br]Solution:[br]Let, [math] f(x) = x^3 – 3x^2 – 10x + 24 [/math][br]Possible factors of 24 are [math] \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 [/math][br][math] \begin{align} \text{Now, } f(2) & = 2^3 -3(2)^2-10(2)+24 \\ & = 8-12-20+24\\ & = 32 - 32 \\ & = 0 \end{align} [/math] [br][math] \therefore x -2 [/math] is a factor of [math] f(x) [/math] [br]Now, using synthetic division, we get,[br][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATsAAACBCAYAAAC/+h1gAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAABD6SURBVHhe7d0LjFTV/QfwLy4oLxEqiC01pYSKGoG0xKapra0FjRW3LW3yj2jFgisVELEWmlhqeAREI0VgifHtFin+a2PRSltpoqVVIUqQl60tUqFSJCAI8lyE5Xa/e34TZoeZ2ZndOzP33vP9JJO558zdx52Z+7vndc9pV1dXF0BEJOHaBY1suyI2bQKmTnXbs2cDAwe6bRGRMJ1hzyIiiaZgJyJeULATES8o2ImIFxTsRMQLCnYi4gUFOxHxgoKdiHhBwU5EvKBgJyJeULATES8o2ImIFxTsRMQLCnYi4gUFOxHxgoKdiHhBwU4SZfNm4KmnLJHhk0+Ahx4CHn0UmDED2LjRXoiaXbvcPzh6NHDXXcD69fZCHtxX8vI62OU7MRKLJ9Lcue5kuuMO4NVX7YX42r7dHdL48cCVVwKrVtkLaRjohg4FLrsMGDsWuOceYOZMYPly2yEq/vlP4IEHgClT3Jfz6quBK64A5syxHbJ4+mmgrs4SkhOnZW+rlSuDYMiQIOjQIQg+85kg+OUv7YUCbNwYBNXV7sHtUnv//SB44IEgGDfO/a81NfaCD/7+9yCYNSsIjh936d273Zvwi1+4dEwdOxYEhw+77WHDsn+mM2cGwQ9/aAnzr38Fwfnnu5+PjFGjTn0+Kffey7UTgmDTJstIw8/w2992r0tebS7ZvfEGsGAB8Oyz7ur58svA0qXA/Pm2Q8T07u1KAKzOXHKJZfqCJYaFC4H333fpXr2AYcOAWbOAAwdcXgydeSbQubMlcmDBh4ea7sILgX37Ila6W7wYuPRSS5ghQ9zzSy+553QPPwzcfrslJJ82B7s773RfpH79XPqii4AlS1wAjKJCTozEGjzYHfxZZ1lGmjOS26JRXw+89172z515K1daIgquvx649lpLZDh+3DbM734HjBhhCWlJm7/hq1cD3bpZwjDg7dxpCYkOXpm2bgX69LGMRhs2uDahrl0tI3nWrHHPHTq453TM+/BDS0TBM88A8+ZZwqR6UoYPd8/EIul//3t6KVByKsnlfMcOV0WQiGOVtn171waRYAcP2kYWPPxjxywRRSyW1tYC48YBgwZZZiNWX9keIwUrSbBjNVafQ4Sx944f0G9/C/z0p64hs8JOngQOHSr8ERb+3UibPNm12bGROYVtd2yAZKSWgoUe7FiqY6n7ttssQ8IVRlS46SZ38vzlL64nKVcbURmtWwdMmFD4g7W4Qp17rm1kceIE0KmTJaLmiSeAw4dd21wKP1MOT+EYGilKO3bJ2nab8Ty8+WZXO+rRwzJbsGkTMHWq2549Gxg40G2Xw1VXAX37Ao89ZhlxsHate4MLxWCW78PgWK4xY4DHHwduucUy4yvbZ8pRAuyT4SiBkSMt07Cpkp2Z991nGVHBLmJ+1tOmWUYjBjnmsWTXsaNlNuJAwxUrgJoa1wiZXgqUU5oGoIRk0iQ3lKsY5R5nly7XmKxEamgIgp/8JAjq6izD/OEPbozWDTdYRrzl+kw5DnT6dEsYviVVVUGwYoVlRMWqVUFQW2sJc/BgECxdaokMixdrnF0BQqnG7t3rSnQ/+pGHY9figm0LDz4I/PznlmFY7KGEjMdhtTRbO9zEicArr1jCMM2RA7xJITJYemMVh3VrVmNTDw5J4bjIfCLfAFlhFvRajRchXjW3bbOMIlWyZPfNbwbBmDGWSDqOyv/iF4PgzTctw7C0x+LNhg2WET/vvhsEY8e6OyTOPjsI+vZ1pbsJE2wHM3GiuxmBJTr+zPDhQfDvf9uLUcE7WnhaZntk3urBOyp44PxceeDf/a77PCWrNrfZXXCBG+6TTSG/udxtdlu2uBsJjhwBXnjBNV6zY4ttOosW2U5JxdtdnnwSqK52Y+v4BvzsZ66B67rrbKdkYwGX37lzznH9MgkeSy0ZQu2gaI1KdlB4iVUdNnCzO7NLF+Caa5o3dosklK5rvmFRhkWaG28Evvc9BTrxhoKdiHhBwU5EvKBgJyJeULATES8o2ImIFxTsRMQLfgY7TkF+//3A9Omxno5cRArnZ7D7z3+A118H3nrLbYtI4qkaKyJe8DPYpS9cwhV4RCTx/Ax26dPcFjrLqIjEmqqxIuIFBTuRqNm1C5gxAxg9GrjrLmD9enshD+4refkZ7FKz81K2xURFKoUzFXPCxSlT3PognEaZcw/OmWM7ZMHV4rhSveTlZ7Dbv982GqnNzhutKTCVHYMaV/9JTZPP+QbvvttNp//22y4vHVf45sLa0iJVY30TizM+fK0pMFXE4sWnr/LPdWOJk65m4mLZXB5NWqRg55PYnPHhK7bAVDFcWCfXOr7pQ6aI68mOGGEJaYmfwS71pfFtjF1szvjwFVtgqhhWSefNs4Thwhk0fLh7Jg6f4uIvmQclOfkZ7FLj7Lp3d8++iM0ZH75iCkyRUl8P1NYC48YBgwZZZiNWX8ePt4QUQtVYn8T2jG+7QgtMoeCiRocOFf7IZ/Jkd0FKX+WfFyYuide+vWVIIfxcXWzBAuDll4HzzgMef9wyPTV3rmvD27Checkh4VhgGjDABbr0OBKKtWuBhQstUYD587OPCuDi2K+95tpXUxgc+Z29807LaPTHP7oDqeypHHl+Bjs20r/6KvDZz5bgmx4jJT3jS4OFJq75W6iuXW0jAzswP/jAtfFH0vLlLmhOm2YZjdjBxDyW7NJXhdu+HVixAqipceNGff5O5+FnsOMf5B/mH+Mf9VXkz/jThVFoylZgipTVq92Bpg8pYYnuxReBkSMtIw0HFY8apZJdC/wJdn/6E7B5s/vm8+Tm1fCCC4Bbb3VXyZ49gcGDbWcPRP6ML41cBaaLLrJEpfGfYTtd5pCSZctc1ZVtdZlSwa6hAU3rAktW/gS73/wG+PWv3fbu3cBHHwGf+pRrtyN+uXy5vzDyZ3xpFFtgqog+fVxpO5tjx5oPl+JwIfbUrlkDbNkCfOtbQL9+p/fESBN/LgOXXGIbjRjgeGKnAh19/eu2kXA847dtax7oeMavW2eJZGIs58W0UydXqE092EHdq5ftFAU7drjqaLZH5rhQDiN65BE34zaXF3j+eQW6PPwp2fHm/xtuaD4JQEqXLsCSJUBVlWUkVGuqSAlRTIFJksmvDgreC8rifqYvfcktvpN0OuPFY361ZvbubRsZfKnCFlNFEkkYv4JdrvsIS1qcFJEo8CvYXXyxbaThwOL0jgoRSSS/gt3nPtd85DldfrltiEiS+RXs2Nv6hS9Ywlx4oW2ISJL5Fewovd2uWze114l4wr9glz64mDfBZ1ZrRSSR/At2DHCpwcOa5VXEG/4FO5bkUr2vX/mKexaRxPMv2BFvluaQk09/2jJEJOn8DHbsgU1vuxORxPMz2DHQffWrlhARH/gZ7D7/eXVOiHim3caNvAs8N94fnm0tkLC88w5w771um8uXZt7RxSn1S/n3RcQP7aqr8we7UuOcg5xmjTifJsf5yim82FRyedtSX+xaUumLXbHv/9Y9W7Hz453o36s/zuvW9nuufbzYc4WEYmeX5+fEUWX5KNiJhGjD9g04duIYep7dE/169rNcKQdO0ty/vyWyaLEay7WTUwvol8LWrcCTT7rtMWNcc1o6Tiy8f78lPFTq978llX7/+fcrefzFvv8KdpXDFVLzle78mqlYpMRqflWD3Qd3Y+jFQzFp6CTLbT0fL/Z79riF0orBtUXylepIwU4kRGEHOwmPn0NPRMQ7CnYi4gUFOxHxgoKdiHhBwU4kRPUn6pueO7Yv8aSw7KZ96CHg0UeBGTOAjRvthQg4cQIYPdotUcop1a66Crj1VuCWW4CRI1366adt5/JRsPPV5s3AU09ZIjnyHdauXS4u8Dzkeunr19sLITpw9EDTc7dOJRwdz0A3dChw2WXA2LHAPfcAM2cCy5fbDhXWvr37EK67zi2+/uyzwGOPAU88ATzzDFBXB0ycCEybZj9QHgp2Ptm+HZg7Fxg/HrjySmDVKnsh3go5LN6lw0GnU6a48/Dqq4ErrgDmzLEd4uT++4G+fV2wI95bxRvMWXpiIIyK1193Y8ky73fr06cib76CnU9693YRgdWfBM3nV8hh8by67z6gc2eXvuYa4O673eQTb7/t8mKDJaNhwyxhOEcjb/WISunu5Engz38GvvY1y8jAq0/XrpYoD6+DXUJrcrnxbunU2Z4ghRzW4sWnz+o1ZIh7fukl9xwL9fXAe+9lP2DmrVxpiZCw/a01Vq92VdjMoEx//Svw7rvArFmWUR7eBbuE1uSkBddfD1x7rSUy8P7X2Fizxj1zOpRMzPvwQ0uEhI2ce/daogivveaeeZKlY6CbMAFYssSdhGXkXbBLaE1OWsB2cc6KkS7VgTl8uHuOhYMHbSMLdgywNBWmO+5wdf1isb2O67ywWs2e14cfdlXtv/0NeOst4MYbbcfy8S7YJbQmJ0VibbC2Fhg3Dhg0yDLjju1k+fCgDx0q7sE77EeNAm6+Gdi5035RC1LtddXVwE03ucdttwHPP+86V/7xD9uxvNRBETf8ImX7UuZ6xES5D2vyZNdmxxJ+rJx7rm1kwfY1BqdcONMGq5DFPjiW77nnXOmsELna61iVYs/xggWWUV5ez3rCsY3swecQoNhYuxZYuNASBZg/P/tUtxE7+HIeFod7sUmpFJ1T31n0nabnkV8e2fQIHYeWnHUWsHSpG6Cbjr2bt9/uup3DxCvDiBHA5ZdbRgtYeuNJ/dFHzWfj5Uy955wD1NRU5nvHYFdJnDyUsyXzwe1yGjYsCGpqLOGbhB58S4f14otBMH26Jcw779hGCKprq5seS99YajklMGTI6QfR0BAEVVVBsGKFZYRk2bIgeO45SxSIJ/PgwZZIw9/DkPPgg5ZRXqrGijdYu9q2rfnAfVaJ162zRFzw7oNXXrGEYZrrGnC0dJjYIfL971uiAKn2umxLlX78sXtmLyGxOM/e2TJRsPMV23daatCOoVyHxTGsbCZhkxarsakHh6T06mU7xQU7CwYPdiOlebBbtrh6/e9/bzuEiJ0LhWDnBe/g4Pgetsu9+aZLp3dqfOMbru2BV5zUAGjmlYna7OLWZtcWPCl4z9SRI8ALL7jGbjYisw1o0SLbKX4KOSzeofTBB247E9vS2UsfhpK32aXj2BmeQGwHSwWZqOMNyhz4zP/1Bz8o6/+sYOdTsJOSK2uwk6J4XY1NaE1OKuSTE6duwu9QleUOB6ko74Idqzw//rFrimD7KNt12bTAHnuRtth/9NQyYD06e7aydQx4F+y43Nojj7g7WDjsh+vWshob4yYrESmA19VYEfGHgp2IeEHBTkS8oGAnIl5QsBMRLyjYiYgXFOxExAsKdiLiBQU7EfFCxYNd+mwTYc08ISKSqeLBbsAANz0PH9wWiasDRw/YFtCtU9p05BIJkajGMsgp0EncHT1+1LaAjh062pZERcXnsxNJik07NmHqMjc54+wRszGwTxknZ4w5To+VPmtMLg0nG7Dv8D40BA2W43Tq0An9z+tvqewU7ERCkh7sOHHnpX0ubdqmvYf2Np2oLdl3ZB+ONxy3VG77j+zHJw2n5s8rBINEIb87F/5v6XP2FYIBrNifaa15/zcvb8BTsBMJSXqwk/JTsBMpk50f78Sk/5+E+uP1llM6nBy02NmQe3TpgTOrWj/koXvn7jizfXE/X+j/yX0KmfC06oyqpuM4o13z7gYe14Dz8zf8K9iJhIgBb8+hPZY6pWfXnk0naku6dyo+oEhhFOxExAu6g0JEvKBgJyJeULATES8o2ImIFxTsRMQLCnYi4gUFOxHxgoKdiHgA+B9siPDvMA6HVQAAAABJRU5ErkJggg==[/img][br][math] \begin{align} \text{Quotient } & = x^2-x-12 \\ & = x^2 -(4-3)x -12 \\ & = x^2 -4x+3x-12 \\ & = x(x-4)+3(x-4) \\ & = (x-4)(x+3) \\ \therefore f(x) & = (x-2)(x-4)(x+3) \\ \text{or, } 0 & = (x-2)(x-4)(x+3) \end{align} [/math] [br][math] \begin{tabular}{|l|l|l| } \hline \text{Either} & \text{Or} & \text{Or}\\[br]x-2=0 & x-4 =0 & x+3 =0 \\[br]\text{or, } x =2 & \text{or, } x=4 & \text{or, } x=-3 \\ \hline \end{tabular} [/math][br][math] \therefore x = 2,4,-3 [/math][br][br]5(f) [math] y^3 + 11y = 6y^2 + 6 [/math][br]Solution:[br]Given,[br][math] y^3+11y=6y^2+6 \\ or, y^3-6y^2+11y-6 = 0 [/math][br][math] \text{Let, } f(y) =y^3-6y^2+11y-6 [/math] [br]Possible factors of 6 are [math] \pm 1, \pm 2, \pm 3, \pm 6 [/math][br][math] \begin{align} \text{Now, } f(1) & = 1^3-6(10^2+11(1)-6 \\ & = 1-6+11-6 \\ & = 0 \\ \end{align} [/math] [br][math] \therefore y-1 [/math] is a factor of [math] f(y) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } &= y^2-5y+6 \\ & = y^2 - 3y -2y + 6 \\ & = y(y-3) -2(y-3) \\ & = (y-3)(y-2) \\ \therefore f(y) & = (y-1)(y-2)(y-3) \\ \text{or, } 0 & = (y-1)(y-2)(y-3) \end{align} [/math][br][br][math] \begin{tabular}{ | l | l | l | } \hline \text{Either} & \text{Or} & \text{Or} \\ y-1 =0 & y-2 = 0 & y-3 =0 \\ \text{or,} y=1 & \text{or, } y = 2 & \text{or,} y =3 \\ \hline \end{tabular} [/math] [br][math] \therefore y = 1,2,3 [/math][br]

1. (a) State factor theorem.[br]Solution:[br]If [math] x-a [/math] is a factor of [math] f(x) [/math] then [math] f(a) = 0 [/math][br]OR[br]If [math] f(a) = 0 [/math] then [math] x-a [/math] is a factor of [math] f(x) [/math].[br][br]1. (b) If [math] (x-a) [/math] is a factor of [math] x^n - a^n [/math], what is the degree of quotient.[br]Solution:[br]The degree of quotient is [math] n - 1 [/math].[br][br]2. In each of the following, use factor theorem to find whether [math] g(x) [/math] is a factor of polynomial [math] f(x) [/math] or not?[br](a) [math] f(x) = x^3 + 9x^2 + 27x + 27; \ \ \ g(x) = x + 3 [/math][br]Solution:[br]Here, [math] f(x) = x^3+9x^2+27x+27 [/math][br]And [math] g(x) = x+ 3 [/math][br]Zero of [math] x+3 [/math] is [math] - 3 [/math][br][math] \begin{align} \text{ Remainder } & = f (-3) \\ & = (-3)^3+9(-3)^2+27(-3) + 27 \\ & = -27 + 81 - 81 + 27 \\ & = 0 \end {align} [/math] [br][math] \therefore \ g(x) [/math] is a factor of polynomial [math] f(x).[/math][br][br](b)[math] f(x) = x^3 + x^2 + 27x + 27, g(x) = x + 3 [/math][br]Solution:[br]Here, [math] f(x) = x^3+x^2+27x+27 [/math][br]And [math] g(x) = x+3 [/math] [br]Zero of [math] x+3 [/math].[br][math] \begin{align} \text{Remainder } & = f(-3) \\ & = (-3)^3+(-3)^2+27(-3)+27\\ & = -27 +9 - 81 +27 \\ & = -72 \neq 0 \end{align} [/math][br][math] \therefore g(x) [/math] is not a factor of polynomial [math] f(x) [/math]. [br][br](c) [math] f(x) = x^3 + 6x^2 + 7x + 9, g(x) = x – 2 [/math][br]Solution:[br]Here, [math] f(x) = x^3+6x^2+7x+9 [/math][br]And [math] g(x) = x-2 [/math][br]Zero of [math] x-2 [/math] is 2.[br][math] \begin{align} \text{Remainder } & = f(2) \\ & = 2^3+6(2)^2+7(2)+9 \\ & = 8 + 24 + 14 + 9 \\ & = 55 \neq 0 \end{align} [/math][br][math] \therefore g(x) [/math] is not a factor of polynomial [math] f(x) [/math]. [br][br](d) [math] f(x) = 3x^3 + x^2 – 20x + 12, g(x) = 3x – 2 [/math][br]Solution:[br]Here, [math] f(x) = 3x^3 + x^2 – 20x + 12 [/math][br]And [math] g(x) = 3x – 2 [/math][br]Zero of [math] 3x – 2 [/math] is [math] \frac{2}{3} [/math].[br][math] \begin{align} \text{Remainder } & = f\left(\frac{2}{3}\right) \\ & = 3\left(\frac{2}{3}\right) ^3+\left(\frac{2}{3}\right) ^2-20\left(\frac{2}{3}\right) +12 \\ & = 3\left(\frac{8}{27}\right) + \frac{4}{9}-\frac{40}{3} +12 \\ & = \frac{8+4-40\times 3 + 12\times 9 }{9} \\ & = \frac{8+4-120+180}{9} \\ & = \frac{0}{9} \\ & = 0 \end{align} [/math][br][math] \therefore g(x) [/math] is a factor of polynomial [math] f(x) [/math]. [br][br](e) [math] f(x) = 8x^3 – 4x^2 + 7x + 9; g(x) = 2x + 1 [/math][br]Solution:[br]Here, [math] f(x) = 8x^3 – 4x^2 + 7x + 9 [/math][br]And [math] g(x) = 2x + 1 [/math][br]Zero of [math] 2x + 1 [/math] is [math] \frac{-1}{2} [/math].[br][math] \begin{align} \text{Remainder } & = f\left(\frac{-1}{2}\right) \\ & = 8\left(\frac{-1}{2}\right)^3-4\left(\frac{-1}{2}\right)^2+7\left(\frac{-1}{2}\right)+9 \\ & = 8\left(\frac{-1}{8}\right) -4\left(\frac{1}{4}\right)-\frac{7}{2}+9 \\ & = -1-1-\frac{7}{2}+9 \\ & = \frac{-2-2-7+18}{2} \\ & = \frac{7}{2} \neq 0 \end{align} [/math][br][math] \therefore g(x) [/math] is not a factor of polynomial [math] f(x) [/math]. [br][br]3. (a) Find the value of [math] k [/math] , if [math] x + 3 [/math] is a factor of [math] 3x^2 + kx + 6 [/math][br]Solution:[br]Let, [math] f(x) = 3x^2+kx+6 [/math] [br]Zero of [math] x+3 [/math] is -3.[br]As [math] x+3 [/math] is a factor of f(x),[br][math] \begin{align} & \text{Remainder } = 0 \\ & \text{ or, } f(-3 ) = 0 \\ & \text{ or, } 3(-3)^2+k(-3)+6=0 \\ & \text{ or, } 27-3k+6=0 \\ & \text{ or, } 33=3k \\ & \text{ or, } k=\frac{33}{11} \\ & \therefore k=11 \end{align} [/math] [br][br][br]3. (b) Find the value of [math] k [/math], if [math] x + 1[/math] is a factor of [math] x^3 – kx^2 – 3x – 6 [/math] [br]Solution:[br]Let, [math] f(x) = x^3 – kx^2 – 3x – 6 [/math] [br]Zero of [math] x+1 [/math] is -1.[br]As [math] x+3 [/math] is a factor of f(x),[br][math] \begin{align} & \text{Remainder } = 0 \\ & \text{ or, } f(-1 ) = 0 \\ & \text{ or, } (-1)^3-k(-1)^2-3(-1)-6 =0 \\ & \text{ or, } -1-k+3-6=0 \\ & \text{ or, } -k-4=0 \\ & \text{ or, } -k=4 \\ & \therefore k=-4 \end{align} [/math] [br][br]3. (c) Find the value of [math] m [/math], for which [math] 2x^4 – 4x^3 + mx^2 + 2x + 1 [/math] is exactly divisible by [math]1 – 2x.[/math][br]Solution:[br]Let, [math] f(x) = 2x^4 – 4x^3 + mx^2 + 2x + 1 [/math] [br]Zero of [math] 1 – 2x [/math] is [math] \frac{1}{2} [/math] .[br]As [math] 1 – 2x [/math] is a factor of f(x),[br][math] \begin{align} & \text{Remainder } = 0 \\ & \text{ or, } f( \frac{1}{2}) = 0 \\ & \text{ or, } 2\left( \frac{1}{2} \right)^4 -4 \left( \frac{1}{2} \right)^3 +m\left( \frac{1}{2} \right)^2 +2\left( \frac{1}{2}\right)+1=0 \\ & \text{ or, } 2\times \frac{1}{16} -4\times \frac{1}{8} +m\times \frac{1}{4}+1+1=0\\ & \text{ or, } \frac{1}{8}-\frac{1}{2}+\frac{m}{4}+2=0 \\ & \text{ or, } \frac{1-4+2m+16}{8} = 0 \\ & \text{ or, } [br] 2m+13 = 0 \\ & \text{ or, } 2m= -13 \\ & \therefore m = - \frac{13}{2} \end{align} [/math] [br][br]4. Factorize the following by using factor theorem.[br](a) [math] 2x^3 + 3x^2 – 3x – 2 [/math] [br]Solution:[br]Let, [math] f(x) = 2x^3 + 3x^2 – 3x – 2 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2 [/math][br][math] \begin{align} \text{Now}, \\ f(1) & = 2(1)^3+3(1)^2 -3(1)-2 \\ & = 2 +3-3 -2 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = 2x^2+5x+2 \\ & = 2x^2 +(4+1)x +2 \\ & = 2x^2 +4x+x+2 \\ & = 2x(x+2) +1 (x+2) \\ & = (x+2)(2x+1) \\ \therefore f(x) & = (x-1) (x+2) (2x+1) \end{align} [/math][br][br]4(b) [math] x^3 + 2x^2 - x -2 [/math][br]Solution:[br]Let, [math] f(x) = x^3 + 2x^2 - x -2 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2 [/math][br][math] \begin{align} \text{Now}, \\ f(1) & = (1)^3+2(1)^2 -1-2 \\ & = 1+2-1-2 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2+3x+2 \\ & = x^2 +(2+1)x +2 \\ & = x^2 + 2x +x +2 \\ & = x(x+2)+1(x+2) \\ & = (x+2)(x+1) \\ \therefore f(x) & = (x-1)(x+2)(x+1) \end{align} [/math][br][br]4(c) [math] y^3 - 6y^2 + 3y + 10 [/math][br]Solution:[br]Let, [math] f(y) = y^3 - 6y^2 + 3y + 10 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2 , \pm 5, \pm 10 [/math][br][math] \begin{align} \text{Now}, \\ f(2) & = 2^3-6(2)^2+3(2)+10 \\ & = 8-24+6+10 \\ & = 0 \end{align} [/math][br][math] \therefore y -2 [/math] is a factor of [math] f(y) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} f(y) & = y^2 - 4y -5 \\ & = y^2 - (5-1)y-5 \\ & = y^2 -5y + y -5 \\ & = y(y-5) +1(y-5) \\ & = (y-5)(y+1) \\ \therefore f(y) & = (y-2)(y-5)(y+1) \end{align}[/math][br][br]4(d) [math] x^3 + 13x^2 + 32x + 20 [/math] [br]Solution:[br]Let, [math] f(x) = x^3 + 13x^2 + 32x + 20 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 [/math][br][math] \begin{align} \text{Now}, \\ f(-2) & = (-2)^3+13(-2)^2+32(-2)+20 \\ & = -8+52-64+20 \\ & = 0 \end{align} [/math][br][math] \therefore x+2 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2 + 11x+ 10 \\ & = x^2 + 10x + x + 10 \\ & = x(x+10) + 1( x+10) \\ & = (x+10)(x+1) \\ \therefore f(x) & = (x+1)(x+2)(x+10) \end{align} [/math][br][br]4(e) [math] 2x^3 + x^2 – 2x – 1 [/math][br]Solution:[br]Let, [math] f(x) = 2x^3 + x^2 – 2x – 1 [/math][br]Possible factors of 2 are [math] \pm 1, \pm \frac{1}{2} [/math][br][math] \begin{align} \text{Now}, \\ f(1) & = 2(1)^3+1^2-2(1)-2 \\ & = 2+1-2-1 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient} & = 2x^2+3x+1 \\ & = 2x^2 + 2x + x + 1 \\ & = 2x(x+1) + 1(x+1) \\ & = (x+1)(2x+1) \\ \therefore f(x) & = (x-1)(x+1)(2x+1) \end{align} [/math][br][br]4(f) [math] x^3 – 23x^2 + 142x – 120 [/math][br]Solution:[br]Let, [math] f(x) = x^3 – 23x^2 + 142x – 120 [/math][br]Possible factors of 2 are [math] \pm 1, \pm 2, \pm 3, \pm 5 \text{ etc } [/math][br][math] \begin{align} \text{Now}, \\ f(1) & = 1^3-23(1)^2+142(1)-120 \\ & =1 -23 + 142 -120 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2 - 22x + 120 \\ & = x^2 - 12x - 10x +120 \\ & = x(x-12) -10(x-12) \\ & = (x-12)(x-10) \\ \therefore f(x) & = (x-1)(x-10)(x-12) \end{align} [/math][br][br]4(g) [math] (x – 1) (2x^2 + 15x + 15) – 21 [/math] [br]Solution:[br][math] \begin{align} \text{Let, } f(x) & = (x – 1) (2x^2 + 15x + 15) – 21\\ & = 2x^3 +15x^2+15x-2x^2-15x-15-2 \\ & = 2x^3+13x^2-36 \end{align} [/math][br]Possible factors of 36 are [math] \pm 1, \pm 2, \pm 3, \pm 4 \text{ etc } [/math][br][math] \begin{align} \text{Now}, \\ f(-2) & = 2(-2)^3+13(-2)^2-36 \\ & =-16+52-36 \\ & = 0 \end{align} [/math][br][math] \therefore x+2 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = 2x^2+9x-18 \\ & = 2x^2 +12x - 3x - 18 \\ & = 2x(x+6) -3 (x+6) \\ & = (x+6)(2x-3) \\ \therefore f(x) & = (x+2)(x+6)(2x-3) \end{align} [/math][br][br]5. Use factor theorem and solve for [math] x [/math].[br](a) [math] x^3 – 4x^2 – 7x + 10 = 0 [/math] [br]Solution:[br]Let [math] f(x) = x^3-4x^2-7x+10 [/math][br]Possible factors of 10 are [math] \pm 1, \pm 2, \pm 5, \pm 10 [/math][br][math] \begin{align} \text{Now, } f(1) & = 1^3-4(1)^2-7(1)+10 \\ & = 1-4-7+10 \\ & = 0 \end{align} [/math][br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATYAAAB9CAYAAAAoVE+yAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAA+QSURBVHhe7d0JjBTFHgbwDzkXASEcPkUBCQoqQgjyMKIQWCCRI0ESFSSAKArhDpgQJQQ5lTwCaIzcAiryBPUhyKXIpWIiKuIBiAQUNIhgOB+7nL79qJq3zdCz27M7O0f190sm3VU76B7T/676V3VVqb/zIEm+/x4YM8acT54M3HOPORcRSaTr7FFExBkKbCLiHAU2EXGOApuIOEeBTUSco8AmIs5RYBMR5yiwiYhzFNhExDkKbCLiHAU2kVTbuxdYuNAWopw/D7z2GjB3LjB+PPDdd/YLUhA9KypOW7kS2LEDaNkSKFPGVno0bw5Uq2YLyXToEPDOO8D+/cAHHwCdOgHz5tkvWgxq2dnA9OlAixbA5cvAo48CTzwBdOli3yR+FNjEaWPHApMm2UKUm28GfvoJqFTJViQTg9bFi0DFikCHDkC9etcGtokTTWvuzTdtRR6W27QBfv0VKFfOVko0dUXDpF8/exIex48Dq1cDH3989at/f2Dx4hQFNWJQYlAryKJFQPv2tmDdcYf5oT780FaIn9AEtoLSGKHAuz4vlJC56SbTy2N8iLwaNgTq1Lk2ZqSV3FzTTfULfqzbvNkWxI/TgY1pjGnTgEGDgLZtgW3b7BfC5uhRYOlSWwiX0aPticfUqcBzz9lCutq+3RzLljVHL9bxbyoxBQ5sZ84AdevaQoa48UYT1DiodNddtjKMZs8GhgyxhXCJHjBgUBs40H8gIa2cPm1PfPCbP3fOFsRPoMDGoMaBmIMHbUWGCJLGcN777wMPP2wL4Xb4MPD550DjxrYiU3F0VApUaGDbsgVo3dq/SS9pjknm335z4Eo21zJvsEFffmbMANq1s4V0V726PfHB0dSsLFsQP4UGtoceAl56yUyjkQzDLij74g7gXLTBg4O/GNO9GBg5eMRBxYzQrJk55uSYoxfrbr3VFsRPofPY9uwBGjUy56VKAcWZ9ZbKeWyxpgqlNV6NZ8/aQgDeuQvr1pm7fuSOtGYN0Llz8f6AGWzDBvMZ2L07//OcNmJ9OO+9F+jaFRg3zlbk4WeCORb+PTt2tJUSrdAWW9p9CMKkqM0U9sV4R1Iz+/++/dYcOc0j7bBr6Zc3GzoU2LjRFiyWeVEqqBUoricP1GLLEEuWmBZbhQq2Ig/nvqxfb2amcroAh4pDhHOTOY2Pg4lpMWF/3z7gX/8yLXI+UsXWNSfWlS8PvPqqfVOeYcPMZDwmuTmvbcQI4JVXgPr17RvEjwJbWHCCbp8+oe2K9uoFvP02cOFCBkz1iMYH33nx3HCDmW18XeBZWqGl31DY+HV5QoADYKtWZWBQoyZNTGTmg+8KaoGE5rcUK43hvB9+AAYMMHMdKlcGuncHRo60XwwPDiJqQYzwcLorGjSNISJuCU2OTUTCI66uaEjzziKSYZSJFBHnKLCJiHMU2ETEOQpsIuIcBTYRcY4Cm4g4x/3AduqUWQ/6hRfMuYg4z/3Axv0XuR70N9+YcxFxnrqiIuIc9wMb16mJ0M7Zko4K2vSWO8Zz7by5c4Hx480SRlIo9wObd/H7atXsiUiKBdn0lkEtO9ushPzMM8DYscCECdoFPgB1RUVSIcimtxz04uqokSXeuRbblCnA00+boCcxKbCJ844cMY0j9uS40vann9ovpFKQTW+5ljnX2fLiNlvshajVViD3A5v3zsa1/iVUdu0C5s83WwVwsyf25nr0MMe0lptr9jjwC36s27zZFsSP+4HtxAl7kiesObaCktOO40Kj3Pvk4EFTrlnTNIImTUrzaY3bt5uj382YdUeP2oL4UVfUVUGS0yHQtKlp4HDV5GhpvX3A6dP2xAc3buB2WxKTApurgiSnQ4Bd0AMHgNq1bUWenTuB1q2v3l+6SLiJBvdwDfpKlJBuyBMP9wNbZB5b2OawBUlOhxC7pWzwLFtmK4qjqBtaB8ENOmLhzkRZWbYgftwPbJEPU9Wq5iihxG1V2YBdvhwYNco0aIuteXNg8eLgr3hyvM2amWNOjjl6sY7bbklM6opKRihur693b9Mr37QJmDnT7Duc1tjiZuDkwI8XfxEcMW3XzlaIHwU2yQiJ6vWxGzpwILB2LbBgga1MtVib3g4dCmzcaAsWy40aAR072grxE9f2e8WVku33ON7PGZm33GJu2WHUoYOZwT5vnq0IB8aKZ581I6N9+9rKPGvWAJ07A48/DixZYiuTLeimt5xRfNNNwOjRZl4bR0OYKKxf375B/LjfYovMY9NzoqHD58W5Af7zz9sKKzJnO6VjKw0aAHPmmOQfJ9Rx6JY3nuidvBnEGIWXLgX27AFWrlRQC8DNwMZ+xssvA2+8Afz4I/D77+a4ejXwySdmvF+c17ixycGvWGErrK1bgdKlTU8vIzRpAvTqBXTpknfFKnsUhJu/Jd4BGcDefRf44w8z2ZFH3iEZ8LjopDiP+bRZs8wjVXy0kh8LNpDY+GGwY7wQN7kZ2LwTUmvVMslWHiMefNCehESs5HQItGxpghsbOqtWAZUrm14fGz/iLjcDW8OGsSfkXn89cNtttuAwJqcHDDDzHL7+2oymcbmbIUPsG8KDQY3TO9ib69YNqFDBfkGc5WZgY1CrU8cWojDoMcHiuqDJaREHuRnYKNbU8rB1Q0VCyN3AxiExP0mZPCciqeRuYLvzTnviwUm63kEEEXGSu4Gtbt1rs8StWtkTEXGZu4GNAwS3324LFteLFxHnuRvYyJtnq1JF+TWRkHA7sHkn6nKahyYwiYSC24HNO2ct1iipiDjH7cDGFlpkFPS++8xRRJzndmAjLvHCaR5c00pEQsH9wMaR0BDv0iQSRu4HNga1+++3BREJA/cDG1fy0MCBSKiU+u67/D0PuChGSa6gvXs3MGWKOedyzdFPPXHnfq3gLSLFVapr1+Rt5sLVc7hsO3HtR86ZlXy8saRy+9OSvrEVJtU3tnh//weOHcDhk4fRoGYD1KpS/GeQw3hjr1Ejr9sYZ7+RfyfO5CqIAptIEe08tBPnLp5Djco1UL+GNlhJpunTzZKDsVzVFb1wIb5d+OPFtQ5ff92cP/nktQvZcvegyKZSYVTSv//CpPr3z/9/Kn/+eH//Cmypw50LC2q1ub+vqEgJ6b+4P/48/Sey78zG8OzhtrbownhjP3YMuHTJFgLKyiq4tUYKbCJFlOjAJonj/nQPEQkdBTaRdMb+6WuvAXPnAuPHm+3t0wW3dezXz+wjwueyO3QwO6E99RTQs6cpczOhFFBgE0lXDGrZ2UCLFsAzzwBjxwITJpjdn9MBd6ReuNBs0nruHLBsmdkJbcECsyv1okVmu/1x4+w/SB4FNnHC3r3mGvOTzo2eAk2dCtSrZwIbccIXZ7izVcQfKl18/rlJmEdPwqtdG2jdGnjxRVuRPApsLjtyBJg2zVzNw4YBn35qv+CGQ4fMjzdoENC2LbBtm/2CR0k2enIv5l45VihTQguYssXTvr0tWFzUgXNS0qXVdvky8NFHwAMP2IoonLhaqZItJI8Cm6t27QLmzwdGjDBdAV7RPXqYoyO4dSyDGltjsRZwKclGz6mcU1eOVbJKYKZ5bl7Q3L8fqFjRVniwbvNmW0gQ5suK4osvTDc0OgDTli3Azz8DkybZiuRRYHMVZzC+8gpw8KAp16xpPnz8kPEREAfw0Rq/694rExo9vrZvN0c+ZxWNdUeP2kKCsFX/11+2EIfPPjNHNpm9GNQGDwbeesvcfZJMgc1VTZuaq758eVvhEe/DeRkq2Y2ehDp92p74YNKeraREYqqCK1PEi/k1LuTKuwRHQGfPNneOrVuBb74BevWyb0yu0AS2gpLLTmIXlM+wMYEbsXOnSeamIOeRCslu9CQN81oFYUQ/cya+F6fz9+kD9O0LHD5s/0OFiOTXunYFevc2r4EDgRUrTA6A6ZAUcTqwBUkuhwa7pbzTc0g+JJLd6Emo6tXtiQ/mwxiIYuFjPewGxvvisPF775lWVxCx8mtMeLJX8PLLtiL5nA5sQZLLzmP3gL+E5cuBUaPMLyWN8Kbv14CI9UqUwho9KdesmTnm5JijF+tuvdUWfEycCCxeHP+Ledj1602+LQjm17gLXHRgYw6XdxXePVLE6cAWJLmc1hJx1bN7wMi+aRMwcybQqZP9QnrYscO/8RDrFc/qG8Vp9KQcP7zNm5scihc/E+xqtmtnKxKE3Ucuod+qla0IgPk1rk4dvf7Yhg3mePfd5pgCocmxZaREXvW8ezL/sXatmRmeJnjt+jUeYr3iWYixOI2etMBZ+xs32oLFMhcz7NjRViQIW1jdu9tCAJH8mt9+IidPmmOkd/D112aUNIkU2NJZUa96fuhGjjR1XpEPWvTF4qhkN3oSjol8jm5z5j6/6X37TKt75Ur7hgRiyz4IDixwEiBb/syjffmlKXsHHNq0MZ/FX37Jn1fDuiRSYHMRnxmaMePa4fvIjNSM7p/7Y9eS1360ZDZ6SgQHfTp3Ns9echY/gxr3yk0V7s/L50HXrQPOngW++sqUvfv28vvjBic8sluagknhCmwuYt6D/TDmTbw4t4jJXl7tDmADZsAA09hgb4cBi42HIUPsG/Iks9FTYpo0MfPB+LB5psxBZO/gsceARx5JyfeswOYi5tNmzTKPVLEbwFEqjo7yrs9gxwvFAVxFdc4c86PxR+S0PTYeXn3VvsFKt0aPlDwFNle1bGmCG++Wq1YBlSubK593/RDKxEaPFJ3+xC7jFcwkL6/obt3MYoAiIRCawBYruSxSFOcv5i8NUra0zzNbklJOB7YgyWWRojiRk7+dVLWKIdvlOAM4HdiCJpdFxC3KsYmIcxTYRMQ5Cmwi4hwFNhFxjgKbiDhHgU1EnKPAJiLOUWATEecosImIc5Ia2LiiaYT3XEQkkZIa2Bo2NBuU88VzkUx1Kid/N/0qWVGbmUjKJb0ryoCmoCaZLudC/g4xFcpqOah0U+rvPPZcRAL6/vfvMeY/Y66cT354Mu6pfc+Vcykcl3zyro4Sy6XLl3D8v8dx6e9LtsbIKpuFBrUa2JI/BTaRIvAGtp7/7InGtRtfOae/zvx15aIszPGzx3Hh0gVbiu3E2RM4fyl//bcgGBCC/Ldj4ffmXXMuCAareP9NUU1/dHqBwU2BTaQIvIFNkk+BTaQEHD55GMP/PRy5F3JtTcnhQpbxrtJb7fpqKFe66FMPqlasinJl4vv3Qb9PvifI4pylryt95ee4rtTVQwH8uRr+o+BEvQKbSBExuB07c8yW8tWoVOPKRVmYqlnxBw8JRoFNRJyjJw9ExDkKbCLiHAU2EXGOApuIOEeBTUSco8AmIs5RYBMR5yiwiYhjgP8B8NUlipXtcdYAAAAASUVORK5CYII=[/img][br][math] \begin{align} \text{Quotient } & = x^2 - 3x -10 \\ & = x^2 -5x + 2x - 10 \\ & = x(x-5) + 2(x-5) \\ & = (x-5)(x+2) \\ \therefore f(x) & = (x-1)(x-5)(x+2) \\ \text{or, } 0 & = (x-1)(x-5)(x+2) \end{align} [/math][br][math][br]\begin{tabular}{|l |l |l | }[br]\hline[br]\text{Either} & \text{Or} & \text{Or} \\[br]x-1=0 & x-5 = 0 & x+2 =0 \\[br]\text{or, } x =1 & \text{or, } x=5 & \text{or, } x= -2 \\[br]\hline[br]\end{tabular}[br][/math][br][math] \therefore x= 1, 5, -2 [/math][br][br]5(b) [math] x^3 + 4x^2 + x – 6 = 0 [/math][br]Solution:[br]Let [math] f(x) = x^3+4x^2+x-6 [/math] [br]Possible factors of 6 are [math] \pm 1, \pm 2, \pm 3, \pm 6 [/math][br][math] \begin{align} \text{ Now, } f(1) & = 1^3+4(1)^2+1-6 \\ & = 1+4+1 -6 \\ & = 6-6 \\ & = 0 \end{align} [/math] [br][math] \therefore x-1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{ Quotient } & = x^2 +5x+6 \\ & = x^2 +3x+2x+6 \\ & = x(x+3) +2 (x+3) \\ & = (x+3)(x+2) \\ \therefore f(x) & = (x-1)(x+2) (x+3) \\ \text{or, } 0 & = (x-1)(x+2)(x+3) \end{align} [/math][br][math] \begin{tabular} { | l | l | l | } \hline \text{ Either } & \text{Or } & \text { Or } \\ x-1 =0 & x+2= 0 & x+3 = 0 \\ \text {or, } x=1 & \text{or, } x=-2 & \text {or, } x = -3 \\ \hline \end{tabular} [/math][br][math] \therefore x = 1, -2, -3 [/math][br][br]5(c) [math] 3x^3 – x^2 – 3x + 1 = 0 [/math][br]Solution:[br]Let [math] f(x) = 3x^3-x^2-3x+1 [/math][br]Possible factors are [math] \pm 1, \pm \frac{1}{3} [/math][br][math] \begin{align} \text{Now, } f(1) &= 3(1)^3-1^2-3(1)+1 \\ & = 3-1-3+1 \\ & = 0 \end{align} [/math][br][math] \therefore x - 1 [/math] is a factor of [math] f(x) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT4AAAB/CAYAAAB2S65NAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAA7cSURBVHhe7d0HjFTVHgbwD5YOIoTiU8wDDYoNCDE8jaAGKUF0jSU+QYNKU1ApoUYJQQwIBEKPWAAF2wuWBzaIUVBq0FAWFI0SQRAQhSy9w7z9OGfejusuzOzO7tx7/98vmdy5Z0fc2Zn55rR7TrlYHmTYpk3A8OHu/pgxQNOm7r6ISGko748iImYo+ETEHAWfiJij4BMRcxR8ImKOgk9EzFHwiYg5Cj4RMUfBJyLmKPhExBwFn4iYo+ATEXMUfCJijoJPRMxR8ImIOWaDb88eYOJEYNQooF8/YPly/wORIPnpJ+D11/2JpIvJ4Nu8GZg1CxgwABg5EhgxAujc2R1FMm7HDvet/NRTQJs2wKpV/geSLiaDb8IEYNo0YPt2d16vHtCuHTB6NHDwoCsTyZhLLnGh99JLwHXX+UJJpxIH3+HDQMOG/iQkmjcHqlUDKlf2BQnKq9dTMq1SJfcGlVJToo85Q+/xx/NrTmHBJu7WrUCDBr4gT04OcNttQI0avsCCiPUfqd9WklXs4Pv6axcUw4b5ghBjs7dCBWD+fF8QZRHtP1K/raSi2MF3553AuHFAy5a+IITefNN9/t97Dxg0yHWtRF5E+4/UbyupKHbwrVsHdOjgT0Kqa1f3+V+6FJgyBejUyf8gyiLaf6R+W0lFsd8S11zj70QAm7m9ewOLFgGzZ/tCCRX120oqzH0Xnj0LDBwIzJ3rC7x4M3fJEneUcDPVbyspMxd8GzcCkycDzz3nC7yTJ91RswjCzWS/raTMXPDdcAPQogWwYIEv8JYtA7KygL59fUGQsdrKuUTJ3kIiHU8rcv22p0+7P4ykVblYHn+/2MqVA0ryr2zaBAwf7u6PGQM0berul5Y1a4A5c4DsbNcHtHAhMHQo8NprwN13+wcF2dq1ri2XLCZA7dr+JEH79kCjRu6JB0C6nlYcpyh27+6mufTo4QvDYMsWN0x99Kh7c9ap44aoOXIzY4Z/kJSEyeAjfokuXgzk5gLVqwMdOwJVqvgfWhGw4CsuvpaDB7uR3cce84V5PvsMuOsu4OGHgbff9oUiecwO9HOKA5tBjzwC3HuvwdCLEPXbSqrSEnwlrzNKRkSk/ygS/bZSpszW+Mxi/9GTT7pRAHaqcf5Or17AM8/4B4QPp63MnOn68j75xF2pwdHdd991YdismX+giJeWPr6SykQfn0SP+m0lWarxSWSo31aSpeATEXMUfCJijoJPRMyxG3wc+hs/Hnj+eS3YJmKM3eD79Vdg5Uq3sCDvi4gZauqKiDl2g+/UKX8nD1clFhEz7AYfZ7nGnW+JDxGJHDV1RcQcBZ9IkEVs7+OgsBt88TWLqGJFf0ckACK693GQ2A2+/fv9nTzq45Mgiejex0Gipq5I0ER07+MgUfCJiDl2gy8+j09z+ETMsRt88Xl8tWq5oyV79rjO81GjgH79gOXL/Q/CjU+LT6lbN7dp/IYN/gciBaipa83mzW6N9gEDgJEjgREjgM6d3THEfvzR7cg4ZIib/dGhg9s6dOxY/4CyxuWgC9sMuKiblCm7S89PnQp8+SVQv74LAitYHeK+i6tXA1de6cq4J+O8ecCBA0DNmq4sZPgUZs92+2/EMfS48xrfX9yQqExFdO/jqLBb44vP47PWx8fNZzliyM2pC+La7SHF3C4Ybjfe6I7ch6PM8X8+d27yN02pKlN2gy8+j8/aG45N3K1bgQYNfEGenBzXLqxRwxeED1vr3G+jMInrUYiQreBbtMg1cVk9+P57YOdOd/z0U9fsZQBYw+YY24fz5/uCcOJWkpMm+ROPG43TXXe5YyhFZO/joLEVfFxpmQH3/vvA778Dhw654yuvuEDkoqRWcONZXh3w3nvAoEHuaoEIOX4cmD4d6NMnhPvqRnDv46CxNbiR+D8qDKsMjRv7EyNYo2jVCqhTxw16RAQzYtcu4MMPfYFIAls1viZNih7M4A7UV1zhTwxhM7d3b9cNwGHRCODTOHJEoSdFsxV8DL1//tOfFMBQzMryJxHFviLO7OUoYqJ4M5dNqpD75BPgt9/+upIT5/iJJLI3qltUX9att/o7Ecbe/smT3eS2RPGpPSG/MJ5TE7dtc/Oy4zg3eP16fyLi2Qu+omaylsms6Qzjc2/RAliwwBd4y5a52m7fvr4gfFirY/9w1aquqRu/cZpLvXr+QSKevSs3fvnFzWVLdPnlbu0zC9asAebMAbKz3dy9hQuBoUPdlQF33+0fFD6clsjBjMKcOKG1KOSv7NX4GjYEqlTxJx5HNa246SZg5sy8Vz7vpf/4Y+Cii9yE5hCHHnFKJr/CC7sp9KQge8HHJt1VV/kT7+qr/R0jGHq8zOGRR4B77/37F4FIxNkLPkrs5+NF+Rb690Tk/2wGX+I+BpzGohqPiCk2gy9xzl6Zr1ckIplmM/hYw+M6fHTzze4oImbYDD7iIpycxnLppb5ARKywG3wcydWepSIm2Q0+ht4tt/gTEbHEbvBxJRYNbIiYVG7jRs5tPz/OfC/NFdp/+AF48UV3n9fPX3utux9XsaK2JBCR9CmXnX3h4CttXBg5vnTQNdeEdqOvUsMvnkxu/1vaX3wXkukvvlT//lv3bsXuA7vRuF5j1K/pZw+UgMUv/rp185qjKbZH+TpxployFHwiaZazIwcnTp9A3Yvq4sq6fgtPKRPJLqKeVFOXu1Tl5vqTUsBr5LlgCHXv/veFkLlcXHxTNItK++9/IZn++/P/n8nnn+rfX8GXOdxUPplan71lqURKWc+5PfHHoT/Q9tq26N+2vy8tPotf/Hv3AmfO+JMkcS3GZLfMUfCJpFm6g0/Sz+50FhExS8EnIuYo+ETEHAWfiJij4BNJs+Onj587VqlQggVuOZTLDbBefRUYNcptDRoUp08D3bq5LVm5xFv79kCvXkCPHkCXLu78zTf9g4NJwSeSZgePHTx3rFm1mDPxGXpt2wItWwJPPAGMGAG88ILbLT0IKlRwO7ZzgypuYTd/vtulj/t5vvsu8MYbbqvSxA2OA0bBJxI048cDjRq54CNeu8WL2VmrYigGxcqVbu5ZwevpuNcnty4dO9YXBI/Z4NuzB5g40bUi+vUDli/3PxDJNNaY2rXzJx7Xj+TlI0Gp9Z09C3z+OdC6tS8ogNeg1qjhT4LHZPBt3gzMmuX2FWdtnC0J7rjPowlMfSY++2kGDgQ2bPA/iIaffnItsVA6ftxtel+tmi9IwLKvvvInacL+uuJYvdo1cwsGNH39NfDzz8Do0b4geEwGH6/nmzYN2L7dnder514/vk5cMCHS+E3MP8CQIS4dOnQIfLMkGTt2uBr8U08BbdoAq1b5H4TNt9+6I5dkKYhlf/7pT9KEX4D79vmTFKxY4Y78Yydi6D39NPDWW+7FCCiTwde8ufvyrFzZFyRIdSmc0GHAjRuXX6Po2BF49lm3EOJ337myELrkEvc540BoqHcUOHTI3ykEBxVYy0on9vPwtU8V+/e4Zw2b3hzBffll1xxftgxYt85tVh9gJoOPTVyuCMM+2LicHFfxCXC3RHrMm/f3ladvvNEdFy92xxDiWmyFtQ4jhf1q58Nm8uHDqd14Zf+jjwKPPQbs3u3/oQuI9+9lZwNdu7pb797AggVuYIZ9SQFnMvgKYrOXX6YclY88dmZ26uRPCuD6S5JZder4O4VgfxyDqihc4YPNzFRvnCv4wQeu1paMovr3WNVmk2nqVF8QYFydJdO4JiAXROWN98vKvHmxWJ8+sVjr1rHYO+/4QosmTOASPbFYTo4vCLd27WKxnj39SQZkT88+d3tnTTHeVCdOuNeisDdk9eqx2LBh/iSNBg2KxVas8CdJGDcuFsvKisUOHPAFHs/5u2fyj58k0zU+1tDZJ7R0KTBlStEVoUhj82j6dKBPH6BZM18oGcM2O7seODSdiM1LvlZ33OEL0oTNU+422KqVL0gC+/fYXVJwqfQvvnDH6693xwBTUzcPm7nsoli0yE0+N2XwYPdB4zdAhvGzXVg3VFG3yOJVD0uW+BOP59yXgaPw6cTBlPvv9ydJiPfvFbY164ED7siRJlq71o3yBpC54OPrxqlrc+f6Ai/+WhV8v0UaU/7IEeDDD31BZq1fX3gXVFG3TC5HX6o40MCpBxyB5xt2yxbXJPnoI/+ANGKzJxkc+OCVI2wWsR/vm2/ceeKAyO23u6s4tm3Ln2zNsgAytwIz5+q2aAFcdhmwc6cvzMMa/333AT17ussOI49vSn4jJ15PyTl+rFWEHK+R5xVfmXod75lxz7ljl391OXcrNi5MwA/HxRfnB07QcXI8J1nzd33ggcD+zuZqfOyaYPAx6BJx+lFWlmtlRB5H5fitnBh6bDuyyiXBwT5XzofjYgBhCD1i0+mhh4AHHwz072wu+NifN3Omu2SNlR5eqcH5l1xUgmEY+f591upYrea0CDZ14zdOc+ElLBHAWR9sIWbCydP5iwhUzCrk6gsJBLObDfGDwfm67IqoXt1dwMClxSKPs7Z37fInBXBuFkcVQ4jdYLwS7+hRYOFCNx2O08x4dc6MGf5BZYCbDHGzIeJGQ9xwSIJHu6yJpJGCLxzMNXVFRBR8ImKOgk9EzFHwiYg5Cj4RMUfBJyLmKPhExBwFn4iYo+ATEXMCEXyJV0mF9IopEQmRQARfkybuOkveeF8krA4ey9+ftGbVAisUS2AEpqnLwFPoSdgdO3XM3wOqVLSw6kU4BWKRApGo2LRzE4b/1624Mea+MWjaQCtuJItLeu0/tt+fFe3M2TPIPZKLM7EzvsSpWrEqGtdv7M/OT8EnkkaJwcfVl29okL+H8b7D+859aC8k92guTp258Faf+4/ux8kz+ev/JYOBkcy/XRT+bolrDiaDYZbqf1Nck/49KanwU/CJpFFi8EnZU/CJZMDuA7vR/z/9cfzUcV9SempXq53yKs+1q9dGpaziT52oVa0WKlVI7b9P9vfkY/jYC8kqn3XueZQv99chCj6vJv9IbqBAwSeSZgy/vYf3+rN8dWvUPfehvZBaVVMPF0mNgk9EzNGVGyJijoJPRMxR8ImIOQo+ETFHwSci5ij4RMQcBZ+ImKPgExFjgP8BiosGhk0+IJMAAAAASUVORK5CYII=[/img][br][math] \begin{align} \text{Quotient} & = 3x^2 +2x-1 \\ & = 3x^2+3x-x-1 \\ & = 3x(x+1) -1(x+1) \\ & = (x+1)(3x-1)\\ \therefore f(x) & = (x-1)(x+1)(3x-1) \\ \text{or, } 0 & = (x-1)(x+1)(3x-1) \end{align} [/math][br][math] \begin{tabular}{ | l | l | l | } \hline \text{Either} & \text{ Or} & \text{Or } \\ x-1 =0 & x+1 = 0 & 3x-1 = 0 \\ \text{or, } x=1 & \text{or, } x=-1 & \text{or, } x =\frac{1}{3}\\ \hline \end{tabular} [/math][br][math] \therefore x= \pm 1, \frac{1}{3} [/math] [br]5(d) [math] x^3-3x^2-9x-5=0 [/math] [br]Solution:[br]Let, [math] f(x) = x^3-3x^2-9x-5=0 [/math][br]Possible factors of 5 are [math] \pm 1, \pm 5 [/math][br][math] \begin{align} f(-1) & = (-1)^3-3(-1)^2-9(-1)-5 \\ & = -1-3+9-5 \\ = & 0 \end{align} [/math] [br][math] \therefore x +1 [/math] is a factor of [math] f(x) [/math][br]Now using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2-4x-5 \\ & = x^2 -5x+x-5 \\ & = x(x-5) + 1(x-5) \\ & = (x-5)(x+1) \\ \therefore f(x) & = (x+1)(x+1)(x-5) \\ \text{or, } 0 & = (x+1)(x+1)(x-5) \end{align} [/math][br][math] \begin{tabular} {|l|l|l|} \hline \text{Either} & \text{Or } & \text{Or } \\ x+1=0 & x+1 =0 & x-5= 0 \\ \text{or, } x= -1 & \text{or, } x= -1 & \text{or, } x= 5 \\ \hline \end{tabular} [/math][br] [br]5(e) [math] x^3 – 3x^2 – 10x + 24 = 0 [/math][br]Solution:[br]Let, [math] f(x) = x^3 – 3x^2 – 10x + 24 [/math][br]Possible factors of 24 are [math] \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 [/math][br][math] \begin{align} \text{Now, } f(2) & = 2^3 -3(2)^2-10(2)+24 \\ & = 8-12-20+24\\ & = 32 - 32 \\ & = 0 \end{align} [/math] [br][math] \therefore x -2 [/math] is a factor of [math] f(x) [/math] [br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } & = x^2-x-12 \\ & = x^2 -(4-3)x -12 \\ & = x^2 -4x+3x-12 \\ & = x(x-4)+3(x-4) \\ & = (x-4)(x+3) \\ \therefore f(x) & = (x-2)(x-4)(x+3) \\ \text{or, } 0 & = (x-2)(x-4)(x+3) \end{align} [/math] [br][math] \begin{tabular}{|l|l|l| } \hline \text{Either} & \text{Or} & \text{Or}\\[br]x-2=0 & x-4 =0 & x+3 =0 \\[br]\text{or, } x =2 & \text{or, } x=4 & \text{or, } x=-3 \\ \hline \end{tabular} [/math][br][math] \therefore x = 2,4,-3 [/math][br][br]5(f) [math] y^3 + 11y = 6y^2 + 6 [/math][br]Solution:[br]Given,[br][math] y^3+11y=6y^2+6 \\ or, y^3-6y^2+11y-6 = 0 [/math][br][math] \text{Let, } f(y) =y^3-6y^2+11y-6 [/math] [br]Possible factors of 6 are [math] \pm 1, \pm 2, \pm 3, \pm 6 [/math][br][math] \begin{align} \text{Now, } f(1) & = 1^3-6(10^2+11(1)-6 \\ & = 1-6+11-6 \\ & = 0 \\ \end{align} [/math] [br][math] \therefore y-1 [/math] is a factor of [math] f(y) [/math][br]Now, using synthetic division, we get,[br][img]data:image/png;base64,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[/img][br][math] \begin{align} \text{Quotient } &= y^2-5y+6 \\ & = y^2 - 3y -2y + 6 \\ & = y(y-3) -2(y-3) \\ & = (y-3)(y-2) \\ \therefore f(y) & = (y-1)(y-2)(y-3) \\ \text{or, } 0 & = (y-1)(y-2)(y-3) \end{align} [/math][br][br][math] \begin{tabular}{ | l | l | l | } \hline \text{Either} & \text{Or} & \text{Or} \\ y-1 =0 & y-2 = 0 & y-3 =0 \\ \text{or,} y=1 & \text{or, } y = 2 & \text{or,} y =3 \\ \hline \end{tabular} [/math] [br][math] \therefore y = 1,2,3 [/math][br]