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Algebra Level pending

( x + 3 ) ( x 2 + a x − 2 ) = x 3 + b x 2 − 6 (x+3)(x^2+ax-2) = x^3 + bx^2 - 6 for some real numbers a a and b . b . What is a ? a ?

3 2 \frac{3}{2} 1 3 \frac{1}{3} 3 3 2 3 \frac{2}{3}

Jason Dyer by Jason Dyer

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1 solution

Jason Dyer Staff
Sep 22, 2016

( x + 3 ) ( x 2 + a x − 2 ) = x 2 ( x + 3 ) + a x ( x + 3 ) − 2 ( x + 3 ) = x 3 + 3 x 2 + a x 2 + 3 a x − 2 x − 6. (x+3)(x^2+ax-2) = x^2(x+3) + ax(x+3) - 2(x+3) = x^3 + 3x^2 + ax^2 + 3ax - 2x - 6 .

We know that the x x terms will cancel, that is, 3 a x − 2 x = 0. 3ax - 2x = 0 . This means 3 a x = 2 x 3ax = 2x so a a must be 2 3 . \frac{2}{3} .

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