Arithmetic Operation ⊗ \otimes

Algebra Level 2

An arithmetic operation ⊗ \otimes is defined as A ⊗ B = A B − ( A + B ) A \otimes B=AB-(A+B) . If P = 8 x , Q = x 2 + 3 x + 2 , P=8x, Q=x^2+3x+2, what is the sum of all the coefficients of the polynomial in x x ( P + Q ) ⊗ P − ( P − 2 Q ) ⊗ P ? (P+Q) \otimes P-(P-2Q) \otimes P?

126 124 122 128

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Mahindra Jain by Mahindra Jain

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1 solution

Will McGlaughlin
Apr 24, 2018

Removing the symbol we reach the equation

( P + Q ) P − ( 2 P + Q ) − ( ( P − 2 Q ) P − ( 2 P − 2 Q ) ) (P+Q)P-(2P+Q)-((P-2Q)P-(2P-2Q))

This expands to;

3 ( P Q − Q ) 3(PQ-Q)

Substituting for P = 8 x P=8x and Q = x 2 + 3 x + 2 Q={ x }^{ 2 }+3x+2

You get the polynomial 24 x 3 + 69 x 2 + 39 x − 6 24{ x }^{ 3 }+69{ x }^{ 2 }+39x-6

24 + 69 + 39 − 6 = 126 24+69+39-6 = \boxed{126}

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