What's the Relationship Between the Zeros, Roots, and Factors of a Polynomial?

Zeros and roots are values that make a polynomial equal to zero. If $x=r$ is a root or zero of $f(x)$, then ((x-r)) is a factor of $f(x)$, and the reverse implication is also true.

The Relationship

For a polynomial $f(x)$, the factor theorem states:

$$f(r)=0 \quad \Longleftrightarrow \quad (x-r) \text{ is a factor of } f(x).$$

The forward implication identifies a factor from a known zero. The reverse implication identifies a zero from a known factor. This connects the equation $f(x)=0$, its solutions, and the polynomial's factorized form.

This follows from polynomial division: dividing $f(x)$ by ((x-r)) leaves constant remainder $f(r)$. Therefore, $f(r)=0$ exactly when the remainder is zero, so the division is exact and ((x-r)) is a factor.

Consider

$$f(x)=x^3-4x^2+x+6.$$

Testing $x=2$ gives

$$f(2)=2^3-4(2^2)+2+6=0.$$

Therefore, the factor theorem confirms that $(x-2)$ is a factor. Polynomial division then gives

$$f(x)=(x-2)(x^2-2x-3)=(x-2)(x-3)(x+1).$$

Hence, the zeros or roots are $x=2$, $x=3$, and $x=-1$. Graphically, these are the $x$-coordinates where the graph of $y$ intersects the -axis.

Once one factor is known, divide to reduce the polynomial's degree and factor the quotient. Finally, check by expanding the factors or substituting every root into the original polynomial; this catches sign and division errors before giving the final set in exact form.

A repeated factor produces a repeated root. For example, $(x-4)^2$ means that $x=4$ has multiplicity $2$. An even-multiplicity root usually makes the graph touch and turn at the axis, while an odd-multiplicity root crosses it.

A common misconception is that root $r$ corresponds to $(x+r)$. The signs are opposite: root $r$ gives factor ((x-r)), while factor $(x+r)$ gives root (-r).

Exam Technique

This is IB Mathematics AA HL content from AHL 2.12. On Paper 1, show $f(r)=0$ before applying the factor theorem; on Paper 2, technology may help identify roots, but your algebraic reasoning and factorization must still be clear.