Know and use the remainder and factor theorems
Factors of Polynomials
What are factors? 🍰
A polynomial can be split into simpler pieces called factors. Think of a polynomial as a big pizza slice. If you can cut it into smaller slices that fit perfectly together, those slices are its factors. When you multiply the factors back together, you get the original polynomial again.
The Remainder Theorem 🔍
If you divide a polynomial $f(x)$ by a linear divisor $(x-a)$, the remainder you get is the value of the polynomial at $x=a$: $$ f(a)=\text{remainder}. $$ So, to find the remainder, simply plug $a$ into $f(x)$.
The Factor Theorem 🎯
If $f(a)=0$, then $(x-a)$ is a factor of $f(x)$. In other words, if plugging $a$ into the polynomial gives zero, you can safely divide by $(x-a)$ without any remainder.
Step‑by‑Step Example
- Choose a simple value for $a$ (often a small integer).
- Compute $f(a)$.
- If $f(a)=0$, write down the factor $(x-a)$.
- Divide $f(x)$ by $(x-a)$ to find the remaining factor(s).
- Repeat if necessary to factor completely.
Example: Factor $f(x)=x^3-4x^2+x+6$.
| Test Value $a$ | $f(a)$ | Factor |
|---|---|---|
| 2 | $2^3-4(2)^2+2+6=0$ | $(x-2)$ |
| -1 | $(-1)^3-4(-1)^2+(-1)+6=0$ | $(x+1)$ |
After dividing by $(x-2)$ and $(x+1)$, the remaining factor is $(x+3)$. So, $$ f(x)=(x-2)(x+1)(x+3). $$
Exam Tips 📚
- Always test small integers first (e.g., -2, -1, 0, 1, 2).
- Remember: if $f(a)=0$, then $(x-a)$ is a factor.
- Use synthetic division to speed up the division process.
- Check your final factors by multiplying them back together.
- Practice with both monic (leading coefficient 1) and non‑monic polynomials.
Revision
Log in to practice.