Basic Polynomial Properties

Algebra Level 2

Let $f(x)$ be some polynomial with real coefficients and $x=p$ as a solution to $f(x)=0$ , and let $g(x)$ be some polynomial with real coefficients which is a factor of $f(x)$ . Which of the following statements must be true?

\frac{f(p)}{g(p)}=0

(x-p)

is a factor of

g(x)

The solutions of

g(x)=0

are also the solutions of

f(x)=0

g(x)

is a polynomial of a higher degree than

f(x)

g'(x)

is a factor of

f'(x)

1 solution

Julian Yu
Jan 2, 2019

If $g(x)$ is a factor of $f(x)$ , then there must exist a polynomial $p(x)$ such that $f(x)=g(x)p(x).$

Therefore, if $\alpha$ is a root of $g(x)$ , $g(\alpha)=0$ which means that $f(\alpha)=0p(\alpha)=0.$

Hence the solutions of $g(x)=0$ are also the solutions of $f(x)=0$ .

Basic Polynomial Properties

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