Divisible Polynomials - Remainder and Factor Theorems

Divisible Polynomials
A polynomial [math]p\left(x\right)[/math] is divisible by a polynomial [math]q\left(x\right)[/math] if there exists a polynomial [math]a\left(x\right)[/math] such that [math]p\left(x\right)=a\left(x\right)\cdot q\left(x\right)[/math], hence the remainder of the division of [math]p\left(x\right)[/math] by [math]q\left(x\right)[/math] is [math]0[/math].
Remainder Theorem
When the divisor polynomial is [math]\left(x-a\right),a\in\mathbb{R}[/math], the [i]Remainder Theorem[/i] allows us to know a priori - that is without calculating the division - the remainder of the division of a polynomial [math]p\left(x\right)[/math] by a binomial [math]\left(x-a\right)[/math].[br]This theorem states that the remainder is [math]r=p\left(a\right)[/math].
Factor Theorem
The [i]Factor Theorem[/i] uses the [i]Remainder Theorem[/i] to provide us with a divisibility criterion for polynomials that can be very useful in applications:[br]If [math]p\left(a\right)=0[/math], then the binomial [math]q\left(x\right)=\left(x-a\right)[/math] is a factor of the polynomial [math]p\left(x\right)[/math]. [br]Conversely, if [math]q\left(x\right)=\left(x-a\right)[/math] is a factor of [math]p\left(x\right)[/math], then [math]p\left(a\right)=0[/math]
Historical Notes
The [i]Factor Theorem[/i] is sometimes referred to as the "[i]Ruffini's Theorem[/i]", as well as the synthetic division algorithm for polynomials is sometimes named "[i]Ruffini's Rule[/i]" because both are the results of the work of the Italian mathematician Paolo Ruffini (1765-1822).
Your turn...
Can you apply the Factor Theorem to decide whether the polynomial [math]p\left(x\right)=5x^3+3x^2+2x+1[/math] is divisible by the binomial [math]q\left(x\right)=x^2+1[/math]?
Can you apply the Factor Theorem to decide whether the polynomial [math]p\left(x\right)=x^3+x^2+6x[/math] is divisible by the binomial [math]q\left(x\right)=x^2+x[/math]?
Close
– Simona Riva

Information: Divisible Polynomials - Remainder and Factor Theorems