Question

if alpha and beta are the zeros of the quadratic polynomial FX equal to x square - 2 X + Q prove that Alpha^/beta^+alpha^=p^^/q^-4p^/q+2

Answer 1

if alpha and beta are zeroes of the quadratic polynomial f x 3x 5x 2 then evaluatealpha beta q2 if one of the zeroes of the cubic polynomial x ax hzek1hdd

Answer 2

Step-by-step explanation:

To prove:

α²/β² + β²/α² = p⁴/q² - 4p²/q + 2

Taking LCM on LHS and RHS, we get:

(α⁴ + β⁴)/(α²β²) = (p⁴ - 4p²q + 2p²) / q² --(i)

Now, we know,

α and β are the two zeroes of the polynomial p(x) = x² - px + q

Therefore, on expressing the polynomial in the form ax² + bx + c,

α + β = -b/a = -p/1 = -p

And,

αβ = c/a = q/1 = q

We know can find the value of α⁴ + β⁴ by squaring α + β tow times as in the attached image and find α²β² and then plug it into equation (i) and then we get LHS = RHS

Refer to the attached images. :D