Algebra II

Polynomial Long Division

Lesson

Polynomial long division works just like long division of numbers — divide, multiply, subtract, bring down — but with polynomial terms.

Step pattern:

Divide the leading term of the dividend by the leading term of the divisor. Write that on top.
Multiply that quotient term by the entire divisor.
Subtract the result from the dividend.
Bring down the next term and repeat.
Stop when the remaining polynomial has degree less than the divisor.

Worked example 1 — clean division (no remainder)

(x^2 + 5x + 6) \div (x + 2)

x² ÷ x = x. So x is the first term of the quotient. Multiply x by (x + 2) to get x² + 2x. Subtract: (x² + 5x) − (x² + 2x) = 3x.

Bring down +6. Now we have 3x + 6. Divide 3x ÷ x = 3. Multiply 3 by (x + 2) to get 3x + 6. Subtract: 0. Done.

\text{Quotient} = x + 3, \quad \text{Remainder} = 0

Worked example 2 — with a remainder

(x^2 + 4x + 7) \div (x + 1)

x² ÷ x = x. Multiply: x² + x. Subtract: 3x. Bring down: 3x + 7. Then 3x ÷ x = 3. Multiply: 3x + 3. Subtract: 4.

\text{Quotient} = x + 3, \quad \text{Remainder} = 4

How to type your answer

Each problem here divides cleanly (no remainder). Type the quotient only, fully simplified. Use ^ for exponents, no spaces, descending order. Examples: x+3, x^2-2x+1, 2x+5.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

(x^2 + 5x + 6) \div (x + 2)

Problem 2

(x^2 + 7x + 12) \div (x + 3)

Problem 3

(x^2 - 3x - 10) \div (x - 5)

Problem 4

(x^2 - 9) \div (x - 3)

Practice

Standard problems matching the lesson.

Problem 5

(x^2 + 8x + 15) \div (x + 3)

Problem 6

(x^2 - 5x + 6) \div (x - 2)

Problem 7

(x^2 + 2x - 8) \div (x - 2)

Problem 8

(x^2 - 16) \div (x + 4)

Problem 9

(2x^2 + 7x + 3) \div (x + 3)

Problem 10

(2x^2 - x - 6) \div (x - 2)

Problem 11

(3x^2 + 5x - 2) \div (x + 2)

Problem 12

(x^3 - 8) \div (x - 2)

Problem 13

(x^3 + x^2 - 4x - 4) \div (x + 1)

Problem 14

(x^3 - 6x^2 + 11x - 6) \div (x - 1)

Problem 15

(x^2 + 4x + 4) \div (x + 2)

Problem 16

(2x^2 + 5x - 12) \div (x + 4)

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

(x^3 + 27) \div (x + 3)

Problem 18

(x^3 + 2x^2 - 5x - 6) \div (x - 2)

Problem 19

(2x^2 + 11x + 12) \div (2x + 3)

Problem 20

(6x^2 - x - 2) \div (2x + 1)

Problem 21

(x^4 - 1) \div (x - 1)

Problem 22

(x^3 - 7x + 6) \div (x - 2)

Practice

Standard problems matching the lesson.

Problem 23

Solid: V = x³ + 2x² − 5x − 6, base (x − 2). Find the other dimension: V ÷ (x − 2).

Problem 24

Rectangle area 2x² + 7x + 3, side (x + 3). Find the other side.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Divide x³ − 4x² + x + 6 by (x − 2). Quotient?

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Quiz

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