What Factor Theorem Do You Know?

Algebra Level 3

P ( x ) = x 3 + 2 x 2 + 2 x + c Q ( x ) = x 2 + b x + b \large P(x) =x^3+2x^2 + 2x + c \qquad\qquad Q(x) = x^2 + bx + b

Consider the polynomials P ( x ) P(x) and Q ( x ) Q(x) above, where b b and c c are constant real numbers and c ≠ 0 c\ne0 .

It is known that Q ( x ) Q(x) is a factor of P ( x ) P(x) . Find the value of b + c b+c .


The answer is 2.

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Akshay Yadav by Akshay Yadav , 20, India

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1 solution

Aareyan Manzoor
May 22, 2016

since Q|P, for some constant a ( x 2 + b x + b ) ( x + a ) = x 3 + 2 x 2 + 2 x + c (x^2+bx+b)(x+a)=x^3+2x^2+2x+c by expanding we have x 3 + ( b + a ) x 2 + ( b + a b ) x + a b = x 3 + 2 x + 2 x + c x^3+(b+a)x^2+(b+ab)x+ab=x^3+2x+2x+c Comparing coefficients we get a b = c ab=c and b + a b = 2 → b + c = 2 b+ab=2\to \boxed{b+c=2} .

we can solve to confirm there exists solutions to the system ( b , c ) = ( 1 , 1 ) , ( 2 , 0 ) (b,c)=(1,1),(2,0)

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Moderator note:

Well written and clearly explained.

For the solutions to the system, a better way to present it is ( b , c ) = ( 1 , 1 ) , ( 2 , 0 ) (b,c) = (1,1), (2,0) . Otherwise, it's unclear if ( 1 , 0 ) (1,0) is a solution.

Calvin Lin Staff - 5 years ago

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Ok i have updated the solution.

Aareyan Manzoor - 5 years ago

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