If - - are the zeros of the polynomial x2-x-12- then form a quadratic equation whose zeros are 2 - and 2 -

F(x) = x2 x 12 is the given polynomial. It is given that, alpha; and beta; are the zeros of the polynomial. #945;+ #946;= ba=–11=1 #160;and #1 ...

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Standard IX

Mathematics

Zeroes of a Polynomial

If α, β are t...

Question

If α, β are the zeros of the polynomial (x² − x − 12), then form a quadratic equation whose zeros are 2α and 2β.

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Solution

f(x) = x² - x - 12 is the given polynomial.
It is given that,α and β are the zeros of the polynomial.
$α + β = \frac{- b}{a} = \frac{- (- 1)}{1} = 1 and αβ = \frac{- 12}{1} = - 12$
Let 2α and 2β be the zeros of the polynomial g(x).
Thus,
Sum of roots = 2α and 2β = 2(α + β) = (2 × 1) = 2
Product of roots = 2α × 2β = 4αβ = 4(-12) = - 48
Now,
The required polynomial g(x) = x² - (Sum of roots)x + Product of roots
= x² - (2)x - 48
= x² - 2x - 48
Hence, the required polynomial is g(x) = x² - 2x - 48.

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If α, β are the zeros of the polynomial (x2 − x − 12), then form a quadratic equation whose zeros are 2α and 2β.

If α, β are the zeros of the polynomial (x² − x − 12), then form a quadratic equation whose zeros are 2α and 2β.