Factor Theorem

The factor theorem connects a polynomial zero P(k)=0 with the linear factor x−k.

Zeros connect polynomial factors to x-intercepts. A smooth curve crosses the horizontal axis at three highlighted and labelled zeros, showing how algebraic roots become geometric x-intercepts.

Evaluating one number can prove that an entire linear expression divides a polynomial.

The theorem

For a polynomial $P(x)$,

P(k)=0\quad\Longleftrightarrow\quad x-k\text{ is a factor of }P(x).

The statement works in both directions. A zero produces a factor, and a factor produces a zero. It links equations, graphs, and algebraic division.

Worked example: factor a cubic

The zeros are $1$, $4$, and $-1$.

Finding possible rational zeros

For a polynomial with integer coefficients, the rational root theorem lists candidates

\pm\frac{\text{factor of constant term}}{\text{factor of leading coefficient}}.

The factor theorem tests those candidates. The rational root theorem proposes possibilities; it does not guarantee that every candidate is a zero.

Solving polynomial equations

Set the polynomial equal to zero, find one factor, divide, and continue until the remaining factors can be solved. The zero-product property then turns the factorization into individual equations.

Always check whether the problem asks for real zeros, rational zeros, or all complex zeros.

Graphical interpretation

The condition $P(k)=0$ means $(k,0)$ is an $x$-intercept. Factorization records the same fact algebraically. A graph can suggest candidate zeros, but exact evaluation establishes them.

Approximate graphing values should not be mistaken for proof when an exact factor is required.

Common mistakes

Testing the wrong sign. For factor $x+3$, use $k=-3$.

Assuming $P(k)=0$ gives factor $x+k$. The matching factor is $x-k$.

Omitting zero coefficients in synthetic division. This shifts every place value.

Stopping after finding one factor. Factor the quotient as far as the question requires.

Treating a rational-root candidate as confirmed. Evaluate it first.