Polynomial Division
A free Precalculus lesson from the “Polynomial Functions” unit, with a worked example and practice problems including step-by-step solutions.
Polynomial division rewrites a rational-looking expression as quotient plus remainder over divisor. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Divide polynomials using long division or synthetic division where appropriate
- Use polynomial division in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. Divide x^2 + 6x + 9 by x + 3. Enter the quotient.
- Worked Example: First identify exactly what the question is asking: Divide x^2 + 6x + 9 by x + 3. Enter the quotient.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Here x + 3 divides evenly: (x + 3)(x + 3) = x^2 + 6x + 9.
- So the quotient is x + 3 with remainder 0.
- Check the result by substituting or estimating: the response should match x + 3 and make sense in the original problem.
Answer: x + 3
Practice problems
1. Divide x^2 + 6x + 9 by x + 3. Enter the quotient.
Show solution
- Warm-up: First identify exactly what the question is asking: Divide x^2 + 6x + 9 by x + 3. Enter the quotient.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Here x + 3 divides evenly: (x + 3)(x + 3) = x^2 + 6x + 9.
- So the quotient is x + 3 with remainder 0.
- Check the result by substituting or estimating: the response should match x + 3 and make sense in the original problem.
Answer: x + 3
2. Divide x^2 + 8x + 16 by x + 4. Enter the quotient.
Show solution
- Warm-up: First identify exactly what the question is asking: Divide x^2 + 8x + 16 by x + 4. Enter the quotient.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Here x + 4 divides evenly: (x + 4)(x + 4) = x^2 + 8x + 16.
- So the quotient is x + 4 with remainder 0.
- Check the result by substituting or estimating: the response should match x + 4 and make sense in the original problem.
Answer: x + 4
3. Use long division to divide x^2 + 10x + 27 by x + 5. Enter the remainder.
Show solution
- x^2 + 10x + 25 = (x + 5)^2 divides evenly.
- The extra + 2 cannot be divided further by x + 5.
- So the quotient is x + 5 and the remainder is 2.
Answer: 2
4. Polynomial long division is needed when:
Choices: the divisor does not divide the dividend evenly · the dividend is a single constant · there are no variables · the divisor equals 1
Show solution
- If a factor cancels cleanly, simplify directly.
- Otherwise long division gives a quotient plus a remainder.
- The remainder has lower degree than the divisor.
Answer: the divisor does not divide the dividend evenly
5. Polynomial long division is needed when: (variation 2)
Choices: the divisor does not divide the dividend evenly · the dividend is a single constant · there are no variables · the divisor equals 1
Show solution
- If a factor cancels cleanly, simplify directly.
- Otherwise long division gives a quotient plus a remainder.
- The remainder has lower degree than the divisor.
Answer: the divisor does not divide the dividend evenly
Practice this interactively with instant feedback and an AI tutor.
Practice Polynomial Division Take the free placement check