Check whether the polynomial f (x) = 4x^3+4x^2 -x -1 is a multiple of 2x +1

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Solution

4x^3+4x^2-x-1=0

If (2x+1) is a multiple of this polinomial.

We can write
(2x+1)(ax^2+bx+c)=0

Than

(2x+1) =0
2x=-1
X=-1/2

That is if x= -1/2 is a solution of this polinomial then we can say it is a multiple of this polinomial

Put x=-1/2 in polinomial

Than
4(-1/2)^3+4(-1/2)^2-x-1=0

-4/8+4/4-1/2+1=0

Than it is a multiple of the polinomial

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