Find a and b such that x+1 and x+2 are factors of the polynomials x^3+ax^2−bx+10.

Joglxym

Joglxym

Answered question

2022-11-18

Find a and b such that x+1 and x+2 are factors of the polynomials x 3 + a x 2 b x + 10

Answer & Explanation

Zoey Benitez

Zoey Benitez

Beginner2022-11-19Added 18 answers

Let x+c be the 3rd/last factor as the expression is of degree three.
So,
( x + c ) ( x + 1 ) ( x + 2 ) = x 3 + a x 2 b x + 10
But,
( x + c ) ( x + 1 ) ( x + 2 ) = ( x + c ) ( x 2 + 3 x + 2 ) = x 3 + x 2 ( c + 3 ) + x ( 3 c + 2 ) + 2 c
So,
x 3 + x 2 ( c + 3 ) + x ( 3 c + 2 ) + 2 c = x 3 + a x 2 b x + 10
Comparing the coefficients of the different powers of x.
2 c = 10 c = 5
a = c + 3 = 5 + 3 = 8
b = 3 c + 2 = 3 5 + 2 = 17 , b = 17

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