Question

if alpha and beta are the zeros of the quadratic polynomial f(x)= ax²+bx+c, then evaluate alpha⁴ +beta⁴

Answer 1

`= \frac{ {b}^(4) + 2 {a}^(2) {c}^(2) - 4a {b}^(2)c }{ {a}^(4) }`

Explanation:

• α + β = -b/a
• αβ = c/a

α⁴ + β⁴

= (α² + β²)² - 2α²β²

= [(α + β)² - 2αβ]² - 2(αβ)²

`= (( - (b)/(a) )^(2) - 2 (c )/(a) ) ^(2) - 2 {( (c)/(a) )}^(2)`

`= { (- (b)/(a)) }^(4) + 4 {( (c)/(a)) }^(2) - 2 {( - (b)/(a) )}^(2) (2 (c)/(a) ) - 2 {( (c)/(a) )}^(2)`

`= { ((b)/(a) )}^(4) + 2 {( (c)/(a)) }^(2) - 4 \frac{ {b}^(2) c}{ {a}^(3) }`

`= \frac{ {b}^(4) + 2 {a}^(2) {c}^(2) - 4a {b}^(2)c }{ {a}^(4) }`