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Question
If If α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3, find a polynomial whose roots are α + 2, β + 2.
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Solution
Since α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3
`alpha+beta="-coefficient of x"/("coefficient of "x^2)`
`=(-(-2))/1`
= 2
Product of the zeroes `="constant term"/("coefficient of "x^2)`
`=3/1`
= 3
Let S and P denote respectively the sums and product of the polynomial whose zeros
α + 2, β + 2
S = (α + 2) + (β + 2)
S = α + β + 2 + 2
S = 2 + 2 + 2
S = 6
P = (α + 2)(β + 2)
P = αβ + 2β + 2α + 4
P = αβ + 2(α + β) + 4
P = 3 + 2(2) + 4
P = 3 + 4 + 4
P = 11
Therefore the required polynomial f(x) is given by
f(x) = k(x2 - Sx + P)
f(x) = k(x2 - 6x + 11)
Hence, the required equation is f(x) = k(x2 - 6x + 11)
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