Question

If α and β are the zeros of the quadratic polynomial f(x) = ax² + bx + c, then evaluate:

(i) α⁴ + β⁴

(ii) \(\frac{1}{a\alpha + b}\) + \(\frac{1}{a\beta + b}\)

(iii)  \(\frac{\beta}{a\alpha + b}\) + \(\frac{\alpha}{a\beta + b}\)

(iv) \(a(\frac{\alpha^2}{\beta}+\frac{\beta^2}{\alpha})\) + \(b(\frac{\alpha}{\beta}+\frac{\beta}{\alpha})\)

Answer 1

(i) Let one root of the given quadratic polynomial is α

Other root of the given quadratic polynomial is β

f(x) = ax² + bx + c

Sum of the roots

Product of the roots

[Using (a + b)² = a² + b² + 2ab]

On substituting values, we get

(ii) Let one root of the given quadratic polynomial is α

Other root of the given quadratic polynomial is β

f(x) = ax² + bx + c

Sum of the roots

Product of the roots

On substituting values, we get

(iii) Let one root of the given quadratic polynomial is α

Other root of the given quadratic polynomial is β

f(x) = ax² + bx + c

Sum of the roots

Product of the roots

(iv) Let one root of the given quadratic polynomial is α

Other root of the given quadratic polynomial is β

f(x) = ax² + bx + c

Sum of the roots

Product of the roots

On substituting values, we get