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Introduction
Dividing Polynomials helps you connect a mathematical rule to a result you can explain and check. Perform leading-term polynomial division, record any remainder, and state divisor restrictions. Include equal-degree and lower-degree divisors. This lesson belongs to exponents and polynomials and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in dividing polynomials.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Use distribution, exponent rules, and integer factors.
Previous lesson: Factoring Quadratic Expressions
Key mathematical vocabulary
- polynomial
- A sum of terms with nonnegative integer variable exponents.
- factor
- An expression multiplied by another to form a product.
- quadratic
- A degree-two polynomial relationship.
- root
- A solution of an equation.
- ratio
- A comparison by division.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Multiply, factor, and divide polynomials
Perform leading-term polynomial division, record any remainder, and state divisor restrictions. Include equal-degree and lower-degree divisors.
Distribution multiplies every term of one factor by every term of the other. Factoring reverses this process. For x²+bx+c, look for two numbers whose sum is b and product is c, then verify by expansion.
For ax²+bx+c with a≠1, find a factor pair of ac whose sum is b, split the middle term, and factor by grouping. If no rational pair works, real linear factors may still exist: use the discriminant to distinguish real-factorable quadratics from those with no real roots. Polynomial long division repeatedly divides leading terms, subtracts the resulting product, and brings down the remainder.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Divide leading terms: x²/x=x. Subtract x(x+12) from the dividend to leave 20x+240.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Multiplying the factors (or divisor and quotient) recovers the polynomial.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Divide x² + 32x + 240 by x+12. State the excluded input.
- Divide leading terms: x²/x=x. Subtract x(x+12) from the dividend to leave 20x+240.
- Divide the remaining leading term: 20x/x=20. Subtract 20(x+12); remainder 0.
- The quotient is x+20. Exclude x=−12, where the original divisor is zero. Equal-degree division can yield a constant plus a remainder/divisor.
Answer: x + 20; x ≠ −12
Verify: Multiplying the factors (or divisor and quotient) recovers the polynomial.
Avoid this mistake: Checking only the product of a proposed factor pair.
Example 2 · Basic
Divide x² + 33x + 260 by x+13. State the excluded input.
- Divide leading terms: x²/x=x. Subtract x(x+13) from the dividend to leave 20x+260.
- Divide the remaining leading term: 20x/x=20. Subtract 20(x+13); remainder 0.
- The quotient is x+20. Exclude x=−13, where the original divisor is zero. Equal-degree division can yield a constant plus a remainder/divisor.
Answer: x + 20; x ≠ −13
Verify: Multiplying the factors (or divisor and quotient) recovers the polynomial.
Avoid this mistake: Canceling terms across addition during division.
Example 3 · Intermediate
Divide (14x² + 20x + 9) by (x² + 1). Give quotient and remainder.
- The leading-term quotient is 14x²/x²=14.
- Multiply: 14(x²+1)=14x²+14. Subtract from the entire dividend.
- The remainder is 20x+-5, of degree less than the divisor. Result: 14+(20x+-5)/(x²+1). The divisor is nonzero for every real x.
Answer: Quotient 14; remainder 20x + -5
Verify: Dividend = divisor·quotient + remainder at x=2, and coefficient comparison establishes the identity.
Avoid this mistake: Forgetting to multiply the factors back to verify all coefficients.
Example 4 · Advanced
Divide (2x + 42) by (x + 21).
- Factor the numerator as 2(x+21).
- Cancel the nonzero common factor x+21.
- The quotient is 2, but x=−21 is excluded from the original expression.
Answer: 2; x ≠ −21
Verify: Multiply the quotient by the divisor to recover the original dividend.
Avoid this mistake: Checking only the product of a proposed factor pair.
Example 5 · Challenge
Divide (3x² + 21x + 9) by (x² + 1). Give quotient and remainder.
- The leading-term quotient is 3x²/x²=3.
- Multiply: 3(x²+1)=3x²+3. Subtract from the entire dividend.
- The remainder is 21x+6, of degree less than the divisor. Result: 3+(21x+6)/(x²+1). The divisor is nonzero for every real x.
Answer: Quotient 3; remainder 21x + 6
Verify: Dividend = divisor·quotient + remainder at x=2, and coefficient comparison establishes the identity.
Avoid this mistake: Canceling terms across addition during division.
Common mistakes and corrections
- Checking only the product of a proposed factor pair.
- Canceling terms across addition during division.
- Forgetting to multiply the factors back to verify all coefficients.
Real-world applications
- Find side lengths from a rectangular area expression. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Recover a missing dimension by dividing area by a known side. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Perform leading-term polynomial division, record any remainder, and state divisor restrictions. Include equal-degree and lower-degree divisors.
Distribution multiplies every term of one factor by every term of the other. Factoring reverses this process. For x²+bx+c, look for two numbers whose sum is b and product is c, then verify by expansion.
Frequently asked questions
What should I know before dividing polynomials?
Use distribution, exponent rules, and integer factors.
How can I check my work?
Multiplying the factors (or divisor and quotient) recovers the polynomial. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Checking only the product of a proposed factor pair. Canceling terms across addition during division. Forgetting to multiply the factors back to verify all coefficients.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Find side lengths from a rectangular area expression.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Perform leading-term polynomial division, record any remainder, and state divisor restrictions. Include equal-degree and lower-degree divisors.
- Checking only the product of a proposed factor pair.
- Canceling terms across addition during division.
- Forgetting to multiply the factors back to verify all coefficients.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials