Algebra
Polynomials
Dividing polynomials
20 practice questions
2 video lessons
Theory + worked examples
Theory
To divide polynomials:
- By a monomial: divide each term separately.
- By a binomial: factor and cancel, or use long division.
- A leftover is the remainder.
Dividing by a monomial splits over each term.
Divide each term by the monomial.
A worked division.
By a monomial:
\[\dfrac{a+b}{c}=\dfrac{a}{c}+\dfrac{b}{c}\]
Factor first when dividing by a binomial.
How to divide
- For a monomial divisor, divide each term.
- For a binomial, factor the numerator.
- Cancel the common factor.
- Use long division if it doesn't factor.
Example 1 — By a monomial
Divide \(\dfrac{6x^2+4x}{2x}\).
Solution
Divide each term.
| \(\dfrac{6x^2}{2x}+\dfrac{4x}{2x}\) | \(=\) | \(3x+2\) |
Example 2 — Simplify by factoring
Divide \(\dfrac{x^2+5x+6}{x+2}\).
Solution
Factor the numerator and cancel.
| \(\dfrac{(x+2)(x+3)}{x+2}\) | \(=\) | \(x+3\) |
Example 3 — Another monomial
Divide \(\dfrac{10x^3-5x^2}{5x^2}\).
Solution
Divide each term.
| \(\dfrac{10x^3}{5x^2}-\dfrac{5x^2}{5x^2}\) | \(=\) | \(2x-1\) |
Example 4 — With a remainder
Divide \(\dfrac{x^2+3x+5}{x+1}\).
Solution
Long division leaves a remainder.
| \(x^2+3x+5\) | \(=\) | \((x+1)(x+2)+3\) |
| \(=\) | \(x+2+\dfrac{3}{x+1}\) |
Common pitfalls
Divide every term by the monomial.
Subtract exponents when dividing variables.
Cancel factors, not individual terms.
Frequently asked questions
How do you divide by a monomial?
Divide each term of the numerator by it.
Simplify \(\dfrac{6x^2}{2x}\).
\(3x\).
How do you divide by a binomial?
Factor and cancel, or use long division.
What is a remainder?
The leftover part that doesn't divide evenly.
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