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Introduction
Subtracting Polynomials helps you connect a mathematical rule to a result you can explain and check. Distribute a subtraction sign to the entire second polynomial before combining terms. 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 subtracting 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
Combine like terms and distribute negative signs.
Previous lesson: Adding Polynomials
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- coefficient
- A factor multiplying a variable term.
- exponent
- A number specifying a power.
- polynomial
- A sum of terms with nonnegative integer variable exponents.
- factor
- An expression multiplied by another to form a product.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Add and subtract polynomials
Distribute a subtraction sign to the entire second polynomial before combining terms.
Add coefficients of like powers while leaving the variable factors unchanged. To subtract a polynomial, first distribute the minus sign to every term in the subtracted polynomial.
Use descending powers or align terms in columns to avoid mixing a squared term with a linear term. Missing powers have coefficient zero. Addition and subtraction do not change the exponents of the terms being combined.
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. Distribute the minus sign to EVERY term in the second polynomial.
- 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. Evaluate both expressions at x=2; their values agree. Grouping like terms proves the identity.
- 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
Subtract: (12x² + 31x + 8) − (x² + 2x + 3).
- Distribute the minus sign to EVERY term in the second polynomial.
- Group the squared terms, linear terms, and constants separately.
- Coefficients become 12−1=11, 31−2=29, and 8−3=5.
Answer: 11x² + 29x + 5
Verify: Evaluate both expressions at x=2; their values agree. Grouping like terms proves the identity.
Avoid this mistake: Changing exponents while adding coefficients.
Example 2 · Basic
Subtract: (13x² + 31x + 8) − (x² + 2x + 3).
- Distribute the minus sign to EVERY term in the second polynomial.
- Group the squared terms, linear terms, and constants separately.
- Coefficients become 13−1=12, 31−2=29, and 8−3=5.
Answer: 12x² + 29x + 5
Verify: Evaluate both expressions at x=2; their values agree. Grouping like terms proves the identity.
Avoid this mistake: Reversing only the first sign during subtraction.
Example 3 · Intermediate
Subtract: (14x² + 31x + 8) − (x² + 2x + 3).
- Distribute the minus sign to EVERY term in the second polynomial.
- Group the squared terms, linear terms, and constants separately.
- Coefficients become 14−1=13, 31−2=29, and 8−3=5.
Answer: 13x² + 29x + 5
Verify: Evaluate both expressions at x=2; their values agree. Grouping like terms proves the identity.
Avoid this mistake: Combining constants with variable terms.
Example 4 · Advanced
Subtract: (2x² + 32x + 8) − (x² + 2x + 3).
- Distribute the minus sign to EVERY term in the second polynomial.
- Group the squared terms, linear terms, and constants separately.
- Coefficients become 2−1=1, 32−2=30, and 8−3=5.
Answer: 1x² + 30x + 5
Verify: Evaluate both expressions at x=2; their values agree. Grouping like terms proves the identity.
Avoid this mistake: Changing exponents while adding coefficients.
Example 5 · Challenge
Subtract: (3x² + 32x + 8) − (x² + 2x + 3).
- Distribute the minus sign to EVERY term in the second polynomial.
- Group the squared terms, linear terms, and constants separately.
- Coefficients become 3−1=2, 32−2=30, and 8−3=5.
Answer: 2x² + 30x + 5
Verify: Evaluate both expressions at x=2; their values agree. Grouping like terms proves the identity.
Avoid this mistake: Reversing only the first sign during subtraction.
Common mistakes and corrections
- Changing exponents while adding coefficients.
- Reversing only the first sign during subtraction.
- Combining constants with variable terms.
Real-world applications
- Combine area expressions for two sections of a garden. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Subtract a cost model from a revenue model. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Distribute a subtraction sign to the entire second polynomial before combining terms.
Add coefficients of like powers while leaving the variable factors unchanged. To subtract a polynomial, first distribute the minus sign to every term in the subtracted polynomial.
Frequently asked questions
What should I know before subtracting polynomials?
Combine like terms and distribute negative signs.
How can I check my work?
Evaluate both expressions at x=2; their values agree. Grouping like terms proves the identity. 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?
Changing exponents while adding coefficients. Reversing only the first sign during subtraction. Combining constants with variable terms.
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: Combine area expressions for two sections of a garden.
- 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
- Distribute a subtraction sign to the entire second polynomial before combining terms.
- Changing exponents while adding coefficients.
- Reversing only the first sign during subtraction.
- Combining constants with variable terms.
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