Question

form a quadratic polynomial whose zeros are 2alpha,2ß if alpha and ß are the zeros of p(x)=x^2+12x+35

Answer 1

The zeroes of the equation p(x) = x^2 + 12x + 35 are α and β as given in the question.

On solving p(x),
= x^2 + 12x + 35
= x^2 + 7x + 5x + 35 = x(x+7) + 5(x+7)
= (x+5) (x+7)
Therefore, the roots of the p(x) are -5 and -7.
Thus, from here α and β are -5 and -7   ....... (1)

And to form the quadratic equation from roots we know,
x^2 + (sum of roots) + (product of roots)

It is given that the roots of the equation we have to form are 2α and 2β, from here

= x^2 + [2α + 2β]x + 2α * 2β
= x^2 + [2(α + β)]x + 4αβ
= x^2 + 2[-5 + (-7)]x + 4(-5 * -7) ........ from (1)
= x^2 + 2(-12)x + 4(35)
= x^2 - 24x + 140

So the equation for the roots given 2α & 2β is x^2 - 24x + 140