When $f(x) = x^3 + 2x^2 + 3x + 2$ is divided by $g(x)$ which is a polynomial with integer coefficient, the quotient and remainder are both $h(x)$ .

Question

So here is the Question :-

When $f(x) = x^3 + 2x^2 + 3x + 2$ is divided by $g(x)$ which is a polynomial with integer coefficient, the quotient and remainder are both $h(x)$ . Given that $h(x)$ is not a constant, find $g$ and $h$.

What I tried :- Since the Quotient and Remainder are both $h(x)$ , we get :- $$f(x) = g(x)h(x) + h(x)$$ $$ \implies x^3 + 2x^2 + 3x + 2 = h(x) [g(x) + 1]$$

Now $f(x)$ can be factorised as $(x^2 + x + 2)(x + 1)$ . So I can conclude that :-

Either $$h(x) = x^2 + x + 2 , g(x) = x$$ Or $$h(x) = x + 1 , g(x) = x^2 + x + 1$$

I am sure I am quite done with the problem , but from here I don't know what to do next . Can Anyone help?

Answer 1

What you are missing to finish is that in a division with remainder we (almost always) require the remainder to be smaller (in some appropriate sense) than the divisor. For division of polynomials with remainder, the appropriate sense is that the degree of the remainder shall be smaller than the degree of the divisor. (If you follow a convention that doesn't assign a negative degree to the zero polynomial, read "the remainder is $0$ or has smaller degree".)

Thus here we have $\deg h < \deg g$, and $h(x) = x+1$, $g(x) = x^2 + x + 1$ is the solution.