Factor Theorem: Difference between revisions

Revision as of 17:49, 15 November 2007

The Factor Theorem is a theorem relating to polynomials

Theorem and Proof

Theorem: If $P(x)$ is a polynomial, then $x-a$ is a factor $P(x)$ iff $P(a)=0$ .

Proof: If $x - a$ is a factor of $P(x)$ , then $P(x) = (x - a)Q(x)$ , where $Q(x)$ is a polynomial with $\deg(Q(x)) = \deg(P(x)) - 1$ . Then $P(a) = (a - a)Q(a) = 0$ .

Now suppose that $P(a) = 0$ .

Apply division algorithm to get $P(x) = (x - a)Q(x) + R(x)$ , where $Q(x)$ is a polynomial with $\deg(Q(x)) = \deg(P(x)) - 1$ and $R(x)$ is the remainder polynomial such that $0\le\deg(R(x)) < \deg(x - a) = 1$ .

This means that $R(x)$ can be at most a constant polynomial.

Substitute $x = a$ and get $P(a) = (a - a)Q(a) + R(a) = 0\Rightarrow R(a) = 0$ .

But $R(x)$ is a constant polynomial and so $R(x) = 0$ for all $x$ .

Therefore, $P(x) = (x - a)Q(x)$ , which shows that $x - a$ is a factor of $P(x)$ .

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Factor Theorem: Difference between revisions

Revision as of 17:49, 15 November 2007

Theorem and Proof

Problems

See also