Concepts · 3 min read
The Remainder Theorem and Factor Theorem Explained
The theorem
You can find a remainder without long division: just substitute. For P(x) = 2x³ − 3x² + 5 divided by (x − 3):
P(3) = 2(27) − 3(9) + 5 = 32, so the remainder is 32.
Computing P(3)
Order-of-operations steps 6 steps
Start: 2 × 3^3 − 3 × 3^2 + 5
- Exponent
3^3 = 27→ 2 × 27 − 3 × 3^2 + 5 - Exponent
3^2 = 9→ 2 × 27 − 3 × 9 + 5 - Multiply
2 × 27 = 54→ 54 − 3 × 9 + 5 - Multiply
3 × 9 = 27→ 54 − 27 + 5 - Subtract
54 − 27 = 27→ 27 + 5 - Add
27 + 5 = 32→ 32
The factor theorem
(x − a) is a factor of P(x) exactly when P(a) = 0. Test the integer divisors of the constant term to hunt for roots.
| a | P(a) for x³ − 4x² + x + 6 | Factor? |
|---|---|---|
| -2 | -20 | no |
| -1 | 0 | (x + 1) |
| 1 | 4 | no |
| 2 | 0 | (x − 2) |
| 3 | 0 | (x − 3) |
Full factorisation
The roots −1, 2 and 3 give x³ − 4x² + x + 6 = (x + 1)(x − 2)(x − 3). Check at x = 4: left = 10, right = 10.
Verify roots on a graph with the graphing calculator.
Where it is used
It speeds up factoring cubics and checking solutions, and connects to the quadratic formula for the leftover quadratic factor.
Frequently asked questions
Is the remainder theorem the same as synthetic division?
Synthetic division is a fast way to divide, and its final number is the remainder, so the two agree.
What if P(a) is not 0?
Then (x − a) is not a factor and P(a) is the remainder.
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