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Introduction
Distributive Property helps you connect a mathematical rule to a result you can explain and check. Multiply every term of a group by its outside factor, including all signs. Use expansion and reverse expansion to connect equivalent forms. This lesson belongs to algebra foundations 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 distributive property.
- 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
Multiply signed numbers and combine like terms.
Previous lesson: Combining Like Terms
Key mathematical vocabulary
- factor
- An expression multiplied by another to form a product.
- expression
- A mathematical phrase representing a value.
- solution
- A value making the original conditions true.
- input
- A value supplied to a rule.
- output
- The result of a rule at an allowed input.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Distribute and simplify
Multiply every term of a group by its outside factor, including all signs. Use expansion and reverse expansion to connect equivalent forms.
To multiply a sum by a factor, multiply every term inside the parentheses: a(b+c)=ab+ac. A negative factor changes the signs of all multiplied terms. After distributing, combine only like terms.
Distribution follows from repeated addition for whole-number factors and extends to real numbers while preserving arithmetic properties. Factoring reverses distribution: ab+ac=a(b+c). Check a proposed simplification by substituting an easy number, then justify the identity with the property.
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 first factor: 12x + 204.
- 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. Substitution x=3 checks the arithmetic; distribution establishes equivalence for every x.
- 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
Simplify 12(x + 17) − 3(x − 1).
- Distribute the first factor: 12x + 204.
- Distribute −3 to both terms of the second parentheses: −3x + 3.
- Combine variable terms and constants: (12−3)x + (204+3)=9x + 207.
Answer: 9x + 207
Verify: Substitution x=3 checks the arithmetic; distribution establishes equivalence for every x.
Avoid this mistake: Multiplying only the first term.
Example 2 · Basic
Simplify 13(x + 17) − 3(x − 1).
- Distribute the first factor: 13x + 221.
- Distribute −3 to both terms of the second parentheses: −3x + 3.
- Combine variable terms and constants: (13−3)x + (221+3)=10x + 224.
Answer: 10x + 224
Verify: Substitution x=3 checks the arithmetic; distribution establishes equivalence for every x.
Avoid this mistake: Failing to distribute a negative sign to all terms.
Example 3 · Intermediate
Simplify 14(x + 17) − 3(x − 1).
- Distribute the first factor: 14x + 238.
- Distribute −3 to both terms of the second parentheses: −3x + 3.
- Combine variable terms and constants: (14−3)x + (238+3)=11x + 241.
Answer: 11x + 241
Verify: Substitution x=3 checks the arithmetic; distribution establishes equivalence for every x.
Avoid this mistake: Combining x and x² as if they were like terms.
Example 4 · Advanced
Simplify 2(x + 18) − 3(x − 1).
- Distribute the first factor: 2x + 36.
- Distribute −3 to both terms of the second parentheses: −3x + 3.
- Combine variable terms and constants: (2−3)x + (36+3)=-1x + 39.
Answer: -1x + 39
Verify: Substitution x=3 checks the arithmetic; distribution establishes equivalence for every x.
Avoid this mistake: Multiplying only the first term.
Example 5 · Challenge
Simplify 3(x + 18) − 3(x − 1).
- Distribute the first factor: 3x + 54.
- Distribute −3 to both terms of the second parentheses: −3x + 3.
- Combine variable terms and constants: (3−3)x + (54+3)=0x + 57.
Answer: 0x + 57
Verify: Substitution x=3 checks the arithmetic; distribution establishes equivalence for every x.
Avoid this mistake: Failing to distribute a negative sign to all terms.
Common mistakes and corrections
- Multiplying only the first term.
- Failing to distribute a negative sign to all terms.
- Combining x and x² as if they were like terms.
Real-world applications
- Calculate the cost of several identical bundles. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Express a rectangle’s area after splitting its width into parts. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Multiply every term of a group by its outside factor, including all signs. Use expansion and reverse expansion to connect equivalent forms.
To multiply a sum by a factor, multiply every term inside the parentheses: a(b+c)=ab+ac. A negative factor changes the signs of all multiplied terms. After distributing, combine only like terms.
Frequently asked questions
What should I know before distributive property?
Multiply signed numbers and combine like terms.
How can I check my work?
Substitution x=3 checks the arithmetic; distribution establishes equivalence for every x. 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?
Multiplying only the first term. Failing to distribute a negative sign to all terms. Combining x and x² as if they were like 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: Calculate the cost of several identical bundles.
- 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
- Multiply every term of a group by its outside factor, including all signs. Use expansion and reverse expansion to connect equivalent forms.
- Multiplying only the first term.
- Failing to distribute a negative sign to all terms.
- Combining x and x² as if they were like 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.
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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