Dividing Polynomials

Master polynomial division using long division, synthetic division, and factoring techniques.

Advanced30 minLesson
Printable worksheet

Definition

Dividing polynomials is the process of finding how many times one polynomial (the divisor) fits into another polynomial (the dividend).
Dividend÷Divisor=Quotient+RemainderDivisor\text{Dividend} \div \text{Divisor} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}
There are three main methods:
  1. 1.Dividing by a monomial: Divide each term separately
  2. 2.Polynomial long division: Similar to numerical long division
  3. 3.Synthetic division: A shortcut when dividing by (x−c)(x - c)
The Remainder Theorem states: When P(x)P(x) is divided by (x−c)(x - c), the remainder equals P(c)P(c).

Try it now

Simplify: 6x23x\frac{6x^2}{3x}

Worked Examples

Divide: 12x3−8x2+4x4x\frac{12x^3 - 8x^2 + 4x}{4x}
1

Set up the division

Divide each term of the numerator by 4x4x → 12x34x−8x24x+4x4x\frac{12x^3}{4x} - \frac{8x^2}{4x} + \frac{4x}{4x}
2

Divide the first term

12x34x=3x2\frac{12x^3}{4x} = 3x^2
3

Divide the second term

8x24x=2x\frac{8x^2}{4x} = 2x → −2x-2x
4

Divide the third term

4x4x=1\frac{4x}{4x} = 1 → +1+1
5

Combine results

Put all terms together → 3x2−2x+13x^2 - 2x + 1

Common Mistakes

Forgetting to include zero coefficients for missing terms

Why it's wrong: In (x3−8)(x^3 - 8), the x2x^2 and xx terms have coefficient 00 and must be included in synthetic division.

Correct: Write (x3+0x2+0x−8)(x^3 + 0x^2 + 0x - 8) before dividing

Using the wrong sign in synthetic division

Why it's wrong: For (x−2)(x - 2), use c=+2c = +2, not −2-2. The sign is opposite to what appears in the divisor.

Correct: (x−c)(x - c) means use +c+c; (x+c)(x + c) means use −c-c

Incorrect subtraction in long division

Why it's wrong: Students often add instead of subtract when eliminating terms.

Correct: Always subtract the entire product: change all signs before adding

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Long Division of Polynomials | 2 Step-by-Step Examples Explained

by Understand The Math

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Interactive Visual

Polynomial Division

(2x3−5x2+3x−1)÷(x−2)\left(2x^{3} - 5x^{2} + 3x - 1\right) \div \left(x - 2\right)
x−2x - 22x32x^{3}−5x2- 5x^{2}+3x+ 3x−1- 1
Write the dividend in descending powers, with the divisor on the left.

Area model: factoring a trinomial

Arrange x², the x-strips and the unit squares into one rectangle: its sides are the factors.

x2x^2
2x2x
3x3x
66
xx
22
xx
33

Length × width = area

x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
Simplify: 6x23x\frac{6x^2}{3x}

Why It Matters

Polynomial division is essential for:
  • Simplifying expressions: Reduce complex fractions to simpler forms
  • Finding roots: Factor higher-degree polynomials to find zeros
  • Calculus preparation: Integration by partial fractions requires polynomial division
  • Engineering: Signal processing and control systems use polynomial operations
Without polynomial division, we couldn't solve cubic or quartic equations, design filters, or analyze complex systems!

Real World Applications

Engineering: Transfer Functions

Control engineers divide polynomials to analyze system behavior and design stable controllers.

Example:

A system's transfer function x2+3x+2x+1\frac{x^2 + 3x + 2}{x + 1} simplifies to x+2x + 2 after polynomial division.

1Try It Yourself

An engineer needs to simplify x2−4x−2\frac{x^2 - 4}{x - 2} for circuit analysis.

What is the simplified form?

Step 1: Write the mathematical expression

Factor and simplify:

Computer Graphics: Bezier Curves

Polynomial division helps split curves for rendering and animation in video games and CGI.

Example:

Dividing a cubic Bezier curve into smaller segments requires polynomial division at specific parameter values.

2Try It Yourself

A graphics programmer needs to find if (x−3)(x - 3) is a factor of x3−6x2+11x−6x^3 - 6x^2 + 11x - 6.

Is (x−3)(x - 3) a factor? Find the quotient.

Step 1: Write the mathematical expression

Use synthetic division with c=3c = 3:

Key Takeaways

  • 1To divide by a monomial, divide each term of the polynomial separately
  • 2Polynomial long division follows the same process as numerical long division
  • 3Synthetic division is a shortcut for dividing by (x−c)(x - c): use +c+c in the process
  • 4Always include zero coefficients for missing terms in the dividend
  • 5The Remainder Theorem: dividing P(x)P(x) by (x−c)(x - c) gives remainder P(c)P(c)

Frequently Asked Questions

Use synthetic division when the divisor is in the form (x−c)(x - c) with a leading coefficient of 1. For any other divisor (like 2x+32x + 3 or x2+1x^2 + 1), use long division.
Use synthetic division when the divisor is in the form (x−c)(x - c) with a leading coefficient of 1. For any other divisor (like 2x+32x + 3 or x2+1x^2 + 1), use long division.
Multiply: Quotient ×\times Divisor ++ Remainder should equal the original Dividend. For example, if (x2+5x+6)÷(x+2)=x+3(x^2 + 5x + 6) \div (x + 2) = x + 3, then (x+3)(x+2)=x2+5x+6(x + 3)(x + 2) = x^2 + 5x + 6.
Write the answer as: Quotient +RemainderDivisor+ \frac{\text{Remainder}}{\text{Divisor}}. For example: 2x+3+5x−12x + 3 + \frac{5}{x-1}.

Glossary

Dividend
The polynomial being divided (the numerator)
Divisor
The polynomial we are dividing by (the denominator)
Quotient
The result of the division (without the remainder)
Remainder
What is left over after division that cannot be divided further
Synthetic division
A shortcut method for dividing a polynomial by a linear factor (x−c)(x - c)

Formula Card

Division Algorithm

Dividend=Divisor×Quotient+Remainder\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}

Relates dividend, divisor, quotient, and remainder

Remainder Theorem

P(x)÷(x−c) has remainder P(c)P(x) \div (x - c) \text{ has remainder } P(c)

Evaluate P(c) to find the remainder when dividing by (x - c)

Factor Theorem

(x−c) is a factor of P(x) if and only if P(c)=0(x - c) \text{ is a factor of } P(x) \text{ if and only if } P(c) = 0

If P(c) = 0, then (x - c) divides P(x) evenly

Today's challengeOne problem, three tries

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