GeoGebra Book: Roots of polynomials of degree 3,4,5 with symbolic solution

★ The general form of the 3rd degree equation (or [b]Cubic[/b]) is: ax³ + bx² + cx + d = 0. Cubics have 3 roots. They are given by: ☛∙z1 = cbrt((((b / a / 3 - b / a) b / a / 3 + c / a) b / a / 3 - d / a) / 2 + 0ί + sqrt(((-(b / a / 3 - b / a) b / a / 3 - c / a) b / a / 3 + d / a)² / 4 + ((b / a / 3 - b / a) b / a / 3 + c / a - ((-b) / a / 3 + b / a - b / a / 3) b / a / 3)³ / 27 + 0ί)) + ((((-b) / a / 3 + b / a) b / a / 3 - c / a + ((-b) / a / 3 + b / a - b / a / 3) b / a / 3) / 3 + 0ί) / cbrt((((b / a / 3 - b / a) b / a / 3 + c / a) b / a / 3 - d / a) / 2 + 0ί + sqrt(((-(b / a / 3 - b / a) b / a / 3 - c / a) b / a / 3 + d / a)² / 4 + ((b / a / 3 - b / a) b / a / 3 + c / a - ((-b) / a / 3 + b / a - b / a / 3) b / a / 3)³ / 27 + 0ί)) + (-b) / a / 3; ☛z2 = cbrt(1 / 54) cbrt((-2 b³ - 27a² d + 9a b c + 3sqrt(3) a sqrt(4a c³ + 27a² d² - b² c² + 4b³ d - 18a b c d + 0ί)) / a³) (ί / 2 sqrt(3) - 1 / 2) + 0ί - 1 / 3 b / a + 1 / 3 ((-c) / a + 1 / 3 b / 3 b / a / a + 1 / 3 b * 2 / 3 b / a / a + 0ί) / (cbrt(1 / 54) cbrt((-2 b³ - 27a² d + 9a b c + 3sqrt(3) a sqrt(4a c³ + 27a² d² - b² c² + 4b³ d - 18a b c d + 0ί)) / a³) (ί / 2 sqrt(3) - 1 / 2) + 0ί); ☛z3 = cbrt(1 / 54) cbrt((-2 b³ - 27a² d + 9a b c + 3sqrt(3) a sqrt(4a c³ + 27a² d² - b² c² + 4b³ d - 18a b c d + 0ί)) / a³) ((-ί) / 2 sqrt(3) - 1 / 2) + 0ί - 1 / 3 b / a + 1 / 3 ((-c) / a + 1 / 3 b / 3 b / a / a + 1 / 3 b * 2 / 3 b / a / a + 0ί) / (cbrt(1 / 54) cbrt((-2 b³ - 27a² d + 9a b c + 3sqrt(3) a sqrt(4a c³ + 27a² d² - b² c² + 4b³ d - 18a b c d + 0ί)) / a³) ((-ί) / 2 sqrt(3) - 1 / 2) + 0ί); ★ The general form of the 4th degree equation (or [b]Quartic[/b]) is: ax⁴ + bx³ + cx² + dx + e = 0. Quartics have 4 roots. They are given by: ☛z1 = (-b) / (4a) - 1 / 2 sqrt(0ί + b² / (4a²) - 2c / (3a) + 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) - 1 / 2 sqrt(0ί + b² / (2a²) - 4c / (3a) - 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) - 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3) - ((-b³) / a³ + 4b c / a² - 8d / a) / (4sqrt(0ί + b² / (4a²) - 2c / (3a) + 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)))); ☛z2 = (-b) / (4a) - 1 / 2 sqrt(0ί + b² / (4a²) - 2c / (3a) + 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / 2 sqrt(0ί + b² / (2a²) - 4c / (3a) - 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) - 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3) - ((-b³) / a³ + 4b c / a² - 8d / a) / (4sqrt(0ί + b² / (4a²) - 2c / (3a) + 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)))); ☛z3 = (-b) / (4a) + 1 / 2 sqrt(0ί + b² / (4a²) - 2c / (3a) + 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) - 1 / 2 sqrt(0ί + b² / (2a²) - 4c / (3a) - 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) - 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3) + ((-b³) / a³ + 4b c / a² - 8d / a) / (4sqrt(0ί + b² / (4a²) - 2c / (3a) + 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)))); ☛z4 = (-b) / (4a) + 1 / 2 sqrt(0ί + b² / (4a²) - 2c / (3a) + 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / 2 sqrt(0ί + b² / (2a²) - 4c / (3a) - 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) - 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3) + ((-b³) / a³ + 4b c / a² - 8d / a) / (4sqrt(0ί + b² / (4a²) - 2c / (3a) + 2^(1 / 3) (c² - 3b d + 12a e) / (3a (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)) + 1 / (3 * 2^(1 / 3) a) (2c³ - 9b c d + 27a d² + 27b² e - 72a c e + sqrt(0ί - 4(c² - 3b d + 12a e)³ + (2c³ - 9b c d + 27a d² + 27b² e - 72a c e)²))^(1 / 3)))). ★ The general form of the 5th degree equation (or [b]Quintic[/b] ) is: a₅*x⁵+a₄*x⁴+a₃*x³+a₂*x²+a₁x+a₀= 0. Quintics have 5 roots. Because one of the roots of the quintic polynomial is a Real number, let's set it explicitly with a slider: x0. Thus the coefficient a₀ of this fifth degree polynomial is the function p(x0). In the applet, you can use the sliders and corresponding buttons to approximate the desired value of a₀. Using the Ruffini rule, dividing the original polynomial by the binary one, we reduce the order of the polynomial to 4, the solution of which is known in symbolic formulas.

 

Roman Chijner

 
Resource Type
GeoGebra Book
Tags
cardan  cubic  equation  polynomial  polynomial-function  quartic  roots 
Target Group (Age)
3 – 19+
Language
English
 
 
 
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