Question

Find a polynomial whose zeros are square of the zeros of the polynomial 3x square +6x -9

Answer 1

p(x) = 3x² + 6x - 9 = 0

a = 3

b = 6

c = -9

Let the zeroes of p(x) be α and β. So that,

→  α + β  =  - b / a  =  - 6 / 3  =  - 2

→  αβ  =  c / a  =  - 9 / 3  =  - 3

Now we have to find a polynomial  q(x)  whose zeroes are α² and β².

    Sum of zeroes  =  α² + β²

⇒  α² + β²  =  (α + β)² - 2αβ  =  (- 2)² - 2 · (- 3)  =  4 + 6  =  10

Hence the coefficient of x in q(x) will be -10 if the coefficient of x² is 1.

    Product of zeroes  =  α²β²

⇒  α²β²  =  (αβ)²  =  (- 3)²  =  9

Hence the constant term in q(x) will be 9 if the coefficient of x² is 1.

So, with coefficient of x² as 1, we write,

q(x) = x² - 10x + 9

Or, with coefficient of x² as 3, the same as that in p(x), we write,

q(x) = 3x² - 30x + 27