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Question
Find the quadratic polynomial, sum of whose zeroes is `( 5/2 )` and their product is 1. Hence, find the zeroes of the polynomial.
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Solution
Let 𝛼 and 𝛽 be the zeroes of the required polynomial f(x).
Then (𝛼 + 𝛽) =` 5/2 `and 𝛼𝛽 = 1
`∴ f(x)=x^2-(∝+β) x+∝β `
`⇒f(x)=x^2-5/2x+1`
`⇒ f(x)=2x^2-5x+2`
Hence, the required polynomial is `f(x)=2x^2-5x+2`
∴` f(x) = 0 ⇒ 2x^2 – 5x + 2 = 0`
⇒ `2x^2 – (4x + x) + 2 = 0`
⇒` 2x^2 – 4x – x + 2 = 0`
⇒ `2x (x – 2) – 1 (x – 2) = 0`
⇒` (2x – 1) (x – 2) = 0`
⇒ `(2x – 1) = 0 or (x – 2) = 0`
`⇒ x =1/2 or x=2`
So, the zeros of f(x) are `1/2` and 2
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