# How to prove two polynomials have no zeroes in common?

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.

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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 8:27
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 8:31
If G(x) also had a remainder, would it still equal zero? – missiledragon May 18 '13 at 8:41

You have that $P(x) = (x+8)G(x) - 22$. What happens when $G(x)$ is zero?
The roots of $G(x)$ are $1$ and $2$.However $P(1)$ and $P(2)=-22$.Hence they share no common roots.