Master the Factor Theorem — one of the most powerful tools for breaking polynomials apart and finding their zeros.
What Is the Factor Theorem?
It's a shortcut: instead of doing long division, you can just plug in a number to test if something is a factor.
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To test if (x − a) divides evenly into a polynomial, just evaluate p(a). If the result is zero — it's a factor!
Is (x − 2) a factor of p(x) = x² − 5x + 6?
Plug in a = 2: p(2) = 4 − 10 + 6 = 0
✓ Yes! Zero means (x − 2) IS a factor.
A zero of a polynomial is an x-value that makes it equal zero. Every zero a gives you a factor (x − a).
If p(3) = 0, then:
AND
These two facts are always linked!
Instead of grinding through polynomial long division, you can test potential factors in seconds by just evaluating the polynomial at one point.
The Factor Theorem is a special case of the Remainder Theorem. When you divide p(x) by (x − a) and the remainder is zero, you've found a factor — and the Factor Theorem tells you this without doing the division!
On a graph, zeros of a polynomial are where the curve crosses the x-axis. The Factor Theorem connects those crossing points to the factored form.
If a parabola crosses the x-axis at x = 2 and x = −3, then:
The Factor Theorem tells you the factors directly from the graph!
The polynomial = a locked door.
A candidate factor (x − a) = a key you want to try.
Plugging in a: if p(a) = 0, the key fits — it's a factor! If not, try a different key.
The Theorem — Both Directions
The Factor Theorem works both ways. This is called a biconditional — it's a two-way street.
Plug a into the polynomial. If you get zero, you've confirmed (x − a) divides evenly. You found a factor!
If you know (x − a) divides evenly into the polynomial, then a must be a zero — plugging it in always gives zero.
The double arrow (⟺) means BOTH directions are always true at the same time.
Quick Check Examples
p(x) = x² − 4
Is (x − 2) a factor?
Test a = 2:
✓ YES — (x − 2) is a factor!
p(x) = x³ − 8
Is (x − 3) a factor?
Test a = 3:
✗ NO — remainder is 19, not 0.
p(x) = x² − x − 6
If p(3) = 0, what's a factor?
Since p(3) = 9 − 3 − 6 = 0:
Divide out to get: (x − 3)(x + 2)
Zeros: x = 3 and x = −2
How To Use It
Here's the systematic approach for applying the Factor Theorem on any problem.
You're given p(x) and asked if (x − a) is a factor. Pull out the value a. Watch the sign — if the factor is (x + 3), then a = −3.
Substitute a into the polynomial and simplify carefully. Use the order of operations. This is the key step — arithmetic errors here will throw off everything.
If p(a) = 0 → (x − a) IS a factor. If p(a) ≠ 0 → it is NOT a factor. The answer is always one of these two outcomes.
Once you confirm a factor, use synthetic division or polynomial long division to divide it out and find the remaining factor(s). Then factor those further if possible.
Worked Examples
Try It Live
Pick a polynomial, enter any value of a, and see instantly whether (x − a) is a factor.
When hunting for rational zeros, try factors of the constant term first (both positive and negative). For example, if the constant is 6, try ±1, ±2, ±3, ±6. This is the Rational Root Theorem working alongside the Factor Theorem.
The Remainder Connection
The Factor Theorem is actually a special case of the Remainder Theorem. Understanding both deepens your mastery.
The remainder equals p(a). You can find any remainder just by plugging in — no long division needed.
The remainder is zero, meaning (x − a) divides evenly — it's a factor. Zero remainder = perfect division.
Think of the Remainder Theorem as the general rule and the Factor Theorem as the special case when everything works out perfectly. Every time you apply the Factor Theorem, you are secretly using the Remainder Theorem — you just already know the remainder will be zero.
Key Vocabulary
Every term you need to know for A2.A.APR.A.1 — searchable and student-friendly.
Quiz Yourself
Test your mastery of A2.A.APR.A.1 — the Factor Theorem!