Question

if a and b are the zeroes of the quadratic polynomial f(x)=ax²+bx+c , then evaluate a³+b³​

Answer 1

EXPLANATION.

a and b are the zeroes of the quadratic polynomial.

⇒ f(x) = ax² + bx + c.

As we know that,

Sum of the zeroes of the quadratic equation.

⇒ a + b = -b/a. - - - - - (1).

Products of the zeroes of the quadratic equation.

⇒ ab = c/a. - - - - - (2).

To evaluate.

⇒ a³ + b³.

⇒ a³ + b³ = (a + b)[a² - ab + b²].

⇒ a³ + b³ = (a + b)[(a² + b²) - ab].

⇒ a³ + b³ = (a + b)[(a + b)² - 2ab - ab].

⇒ a³ + b³ = (a + b)[(a + b)² - 3ab].

Put the values in the equation, we get.

⇒ a³ + b³ = (-b/a)[(-b/a)² - 3(c/a)].

⇒ a³ + b³ = (-b/a)[b²/a² - 3c/a].

⇒ a³ + b³ = (-b/a)[b² - 3ac/a²].

⇒ a³ + b³ = -b³ + 3abc/a³.

⇒ a³ + b³ = 3abc - b³/a³

                                                                                                                       

MORE INFORMATION.

Conjugate roots.

(1) = If D < 0.

One roots = α + iβ.

Other roots = α - iβ.

(2) = If D > 0.

One roots = α + √β.

Other roots = α - √β.