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If $α$ and $β$ are the zeros of the quadratic polynomial $f (x) = a x 2 + b x + c$ then evaluate $a α 2 β + β 2 α + b α β + β α .$

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-b

answer is B.

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Detailed Solution

Given that

α

and

β

are the zeroes of the quadratic polynomial

f (x) = a x 2 + b x + c .

Then sum and product of roots can be written as

Sum of zeroes = − b a ⇒ α + β = − b a

And,Product = c a ⇒ α β = c a

Given to evaluate a α 2 β + β 2 α + b α β + β α,

a α 2 β + β 2 α + b α β + β α

can be written as,
Use properties:

(x + y) 3 = x 3 + y 3 + 3 x y (x + y)

and

(x + y) 2 = x 2 + y 2 + 2 x y

Substitute

α + β = − b a and α β = c a

into the equation,

a α 2 β + β 2 α

+ b α β + β α = a (α + β) 3 − 3 α β (α + β) α β + b (α + β) 2 − 2 α β α β

= a (− b / a) 3 − 3 (c / a) (− b / a) c / a + b (− b / a) 2 − 2 (c / a) c / a

= a − b 3 a 3 + 3 b c a 2 c / a + b b 2 a 2 − 2 c a c / a

= a − b 3 + 3 a b c a 3 × a c + b b 2 − 2 a c a 2 × a c

⇒ a α 2 β + β 2 α + b α β + β α = − b 3 + 3 a b c a c + b 3 − 2 a b c a c = − b 3 + 3 a b c + b 3 − 2 a b c a c = a b c a c = b

Therefore value of

a α 2 β + β 2 α + b α β + β α is b

.
Hence the correct option is 2.

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