Curriculum fit: US Algebra 2. Aligned to Common Core HSA-APR.B.2 (the remainder theorem) and HSA-APR.B.3 (zeros of polynomials). Check your own state standards and course.
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What Is It?
Dividing a polynomial by x − c is quick with synthetic division, which uses only the coefficients.
The remainder theorem says that when a polynomial f(x) is divided by x − c, the remainder is f(c). So you can find a remainder just by substituting.
The factor theorem follows: x − c is a factor of f(x) exactly when f(c) = 0. Use it to find roots of cubic and higher-degree polynomials.
Key Formulas and Ideas
- Remainder theorem: the remainder when f(x) is divided by x − c is f(c).
- Factor theorem: x − c is a factor of f(x) if and only if f(c) = 0.
- Synthetic division: write the coefficients, bring down the first, multiply by c, add down the column, repeat.
- To divide by x + 2, use c = −2.
Worked Examples
Find All Roots of a Cubic
Solve x³ − 2x² − 5x + 6 = 0.
- Test c = 1: f(1) = 1 − 2 − 5 + 6 = 0, so x − 1 is a factor.
- Synthetic division with 1 on 1, −2, −5, 6 gives 1, −1, −6 and remainder 0, so the quotient is x² − x − 6.
- Factor: x² − x − 6 = (x − 3)(x + 2).
✓ Answer x = 1, x = 3 or x = −2
Remainder Theorem
Find the remainder when 2x³ + 3x² − x + 4 is divided by x + 2.
- x + 2 means c = −2.
- f(−2) = 2(−8) + 3(4) − (−2) + 4.
- = −16 + 12 + 2 + 4 = 2.
✓ Answer Remainder 2
Use the Factor Theorem to Find k
Find k so that x − 2 is a factor of x³ + kx² − x − 10.
- x − 2 is a factor, so f(2) = 0.
- f(2) = 8 + 4k − 2 − 10 = 4k − 4.
- Set 4k − 4 = 0, so k = 1.
✓ Answer k = 1
Common Mistakes to Avoid
- Using c = 2 instead of c = −2 when dividing by x + 2.
- Leaving out a zero coefficient for a missing term, such as x³ + 1, which has coefficients 1, 0, 0, 1.
- Forgetting that the last number in the synthetic division is the remainder, not part of the quotient.
- Stopping after finding one root and not factoring the quotient further.
Practice Questions
Frequently Asked Questions
When can I use synthetic division?
Synthetic division works when you divide by a linear factor of the form x − c. For a divisor of higher degree, use long division.
How do I choose which values of c to test?
For a polynomial with integer coefficients, try factors of the constant term (positive and negative). For x³ − 2x² − 5x + 6, test ±1, ±2, ±3 and ±6.
What does a remainder of zero tell me?
It tells you the divisor is a factor, and that c is a root of the polynomial.
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High SchoolSources and Review
Reviewed by StudyMorrow editorial review on 6 Nov 2026.