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Question
Find the quadratic polynomial, sum of whose zeroes is 0 and their product is -1. Hence, find the zeroes of the polynomial.
Solution
Let 𝛼 and 𝛽 be the zeroes of the required polynomial f(x).
Then (𝛼 + 𝛽) = 0 and 𝛼𝛽 = -1
`∴ F(x)=x^2-(∝+β)x+∝β `
⇒ `f(x)=x^2-o x+(-1)`
⇒`f(x) = x2 ˗ 1`
Hence, required polynomial `f(x) =x^2-1`
`∴ f(x)=0⇒ x^2-1=0`
`⇒ (x+1) (x-1)=0`
`⇒(x+1)=0 or (x-1)=0`
` ⇒ x=-1 or x=1`
So, the zeroes of f(x) are -1 and 1.
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