Polynomials and Rational Functions
4.3 Polynomial Zeroes
Zeroes, Factors, and Graphs
Finding zeroes links algebraic factoring to x-intercepts on graphs.
Theorem: Factor Theorem
For polynomial $P(x)$, $x-c$ is a factor of $P(x)$ if and only if $P(c)=0$.
Theorem: Remainder Theorem
The remainder when dividing $P(x)$ by $x-c$ is $P(c)$.
Example: Test whether a binomial is a factor
Let
$$P(x)=x^3-4x^2-x+4.$$
Is $x-1$ a factor?
Show Solution
Evaluate $P(1)$:
$$P(1)=1-4-1+4=0.$$
Since $P(1)=0$, $x-1$ is a factor.
MyOpenMath: Practice: factor theorem checks (placeholder)
Rational Zero Candidates
The Rational Zero Theorem narrows where we should test.
Theorem: Rational Zero Theorem
If $\frac{p}{q}$ is a rational zero of
$$a_nx^n+\cdots+a_0,$$
then $p$ divides $a_0$ and $q$ divides $a_n$.
Example: Find all rational zeroes
Find zeroes of
$$f(x)=x^3-6x^2+11x-6.$$
Show Solution
Possible rational zeroes: $\pm1,\pm2,\pm3,\pm6$.
Test quickly:
$$f(1)=0,$$
so factor out $(x-1)$ to get
$$x^2-5x+6=(x-2)(x-3).$$
All zeroes are
$$x=1,2,3.$$