Question

if alpha and beta are the zeros of x^2-2x-8 write a quadratic polynomial whose zeroes are 2alpha and 2 beta​

Answer 1

Given :-

x² - 2x - 8

To Find :-

Quadratic polynomial whose zeroes are 2α and 2β

Solution :-

x² - 2x - 8

x² - (4x - 2x) - 8

x² - 4x + 2x - 8

x(x - 4) + 2(x - 4)

(x - 4)(x + 2)

So,

Either

x - 4 = 0

x = 4

or,

x + 2 = 0

x = -2

Now

Sum of zeroes = -b/a

Sum = -(-2)/1

Sum = 2/1

Sum = 2

Product = c/a

Product = -8/1

Product = -8

New quadratic polynomial whose zeroes are 2α and 2β

Sum = 2α + 2β

Sum = 2(α + β)

Sum = 2(2)

Sum = 4

Product = 2α × 2β

Product = 2(αβ)

Product = 2(-8)

Product = -16

Standard form of quadratic polynomial = x² - (α + β)x + αβ

Quadratic polynomial = x² - (4)x + (-16)

Quadratic polynomial = x² - 4x - 16