How to prove two polynomials have no zeroes in common?

Question

The question asked: Divide the polynomial $P(x) = x^3 + 5x^2 - 22x - 6$ by $G(x) = x^2 - 3x + 2$. I did, and got the answer: $(x+8)(x^2-3x+2)-22$.

However, it now asks to: "Show that $P(x)$ and $G(x)$ have no zeros in common."

How do I prove this?

Thanks.

You've just worked out that $P(x) = (x+8)G(x) - 22$ so if $G(x) = 0$ what is $P(x)$? — Tim, May 18 '13 at 08:27
You seemed to have already cracked it. Assume both have root $\alpha$ in common.
This implies $P(\alpha)=G(\alpha)=0$ but you have you have $P(\alpha)=-22$ when $G(\alpha)$ is $0$. This contradicts the fact that $\alpha$ exists. — Inceptio, May 18 '13 at 08:31

score 3 · Accepted Answer · answered May 18 '13 at 08:28

3

You have that $P(x) = (x+8)G(x) - 22$. What happens when $G(x)$ is zero?

answered May 18 '13 at 08:28

N.U.

I have a small question. If G(x) also had a remainder, would it still equal zero? – missiledragon May 18 '13 at 08:38
I didn't understand your question. Could you explain some more? – N.U. May 18 '13 at 09:23
Nevermind, it's not a problem anymore – missiledragon May 18 '13 at 10:15

score 1 · Answer 2 · edited May 18 '13 at 08:32

1

The roots of $G(x)$ are $1$ and $2$.However $P(1)$ and $P(2)=-22$.Hence they share no common roots.

edited May 18 '13 at 08:32

Inceptio

answered May 18 '13 at 08:27

Shaswata

2 Answers2