the polinomial f(x)= x^3+ax^2+bx+c is such that when divided by x^2-4 leaves remainder 2x+11 and has factor x+1. Find the value of a,b and c

Question

A' level question

Answer 1

x^3+ax^2+bx+c-2x-11=(x^2-4)(x+1)=x^3+x^2-4x-4.

There can be no more factors because the highest power of x is 3: x^2 * x=x^3and we know x^2-4 and x+1 are given factors once we subtract the remainder from the dividend.

Therefore a=1 (x^2 term); b-2=-4 (x term), making b=-2; c-11=-4, making c=7.