Polynomial long division works exactly like the numeric kind: divide the leading terms, multiply back, subtract.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The Remainder Theorem
Dividing by (x − a) leaves a remainder of exactly f(a). No division required.
And the Factor Theorem
If f(a) = 0, the remainder is zero, so (x − a) is a factor.
Why it matters
It lets you test a candidate root by substitution rather than by dividing.
Polynomial long division
The procedure mirrors numerical long division: divide the leading terms, multiply back, subtract, bring down. Keeping columns aligned by degree is what makes it work, so include zero coefficients for missing powers.
Synthetic division is the shortcut
When dividing by a linear factor x − a, synthetic division does the same arithmetic with only the coefficients. It is faster and less error-prone, but it applies only to linear divisors, which is worth remembering.
The remainder theorem
Dividing f(x) by x − a leaves remainder f(a). So evaluating a polynomial and dividing by a linear factor are the same computation. A remainder of zero is exactly the factor theorem.
Check by multiplying back
Quotient × divisor + remainder should reproduce the original polynomial. This is the same check as for numerical division and it catches sign errors, which are the usual failure in synthetic division.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Idris Test a Root
Idris checks whether (x − 2) divides f(x) = x³ − 3x² + 4.
- Step 1
The Remainder Theorem says the remainder is f(2).
Step 4: Your Turn
Practice makes it stick.
The Remainder
Problem 1 of 2
f(x) = x² + 1 divided by (x − 3). What is the remainder?
The Factor
Problem 2 of 2
f(4) = 0. Is (x − 4) a factor? 1 for yes, 0 for no.
Remainders and Factors
1 of 8
f(x) = x² − 5, divided by (x − 2). Remainder?
2 of 8
f(x) = x³, divided by (x − 1). Remainder?
3 of 8
f(x) = x² − 9, divided by (x − 3). Remainder?
4 of 8
Is (x − 3) therefore a factor of x² − 9? 1 for yes, 0 for no.
5 of 8
f(x) = 2x + 6, divided by (x + 3). Remainder?
6 of 8
Degree 5 divided by degree 2. What is the degree of the quotient?
7 of 8
f(x) = x² + x, divided by (x − 2). Remainder?
8 of 8
Select every statement that follows from f(a) = 0.
Step 5: Quick Check
Show what you know.
Question 1 of 1
f(x) = x² + 2, divided by (x − 3). Remainder?
What You Learned
- Polynomial long division mirrors the numeric kind.
- The remainder on dividing by (x − a) is simply f(a).
- A zero remainder means (x − a) is a factor and a is a root.