Distributive Property & Factoring
Learn to expand linear expressions with the distributive property (even with negative multipliers) and factor out the GCF, then combine like terms to reach simplest form.
What you'll do in this lesson
A voice-first session with the Crimsora tutor on Distributive Property & Factoring, then targeted practice and FRQs — with the tutor adapting to where you get stuck.
What this lesson covers
You already know how to combine like terms. But what do you do with an expression like ? The parentheses are in the way, and you can't combine anything until they're gone. That's where the distributive property comes in — it lets you multiply across a sum or difference and rewrite the expression without parentheses.
Factoring is the same move played backwards: instead of breaking parentheses open, you pull a common factor out and build parentheses. In this lesson you'll practice both directions, pay close attention to negative multipliers (the single biggest source of mistakes here), and finish by combining like terms so every answer lands in simplest form.
Factoring is the same move played backwards: instead of breaking parentheses open, you pull a common factor out and build parentheses. In this lesson you'll practice both directions, pay close attention to negative multipliers (the single biggest source of mistakes here), and finish by combining like terms so every answer lands in simplest form.
The Distributive Property: Multiply Everything Inside
The distributive property says that multiplying a number by a sum is the same as multiplying it by each part and then adding:The key word is every. The factor outside the parentheses multiplies every single term inside, not just the first one. So , and .
Why is this true? Think about area. A rectangle that is 4 units tall and units wide can be split into a 4-by- piece and a 4-by-7 piece. Total area: . Same rectangle, two ways of describing it — which is exactly what equivalent expressions means.
Subtraction inside parentheses works the same way, because is really . So .
The most common error is distributing to only the first term: writing . A quick check catches it. Let . The original is . The wrong answer gives ; the correct one gives . Substituting a number is your safety net all through this lesson — equivalent expressions must give equal values for every input.
Also watch the case where a variable is outside: is not linear and isn't part of this lesson, but is — distribute to all three terms.
Why is this true? Think about area. A rectangle that is 4 units tall and units wide can be split into a 4-by- piece and a 4-by-7 piece. Total area: . Same rectangle, two ways of describing it — which is exactly what equivalent expressions means.
Subtraction inside parentheses works the same way, because is really . So .
The most common error is distributing to only the first term: writing . A quick check catches it. Let . The original is . The wrong answer gives ; the correct one gives . Substituting a number is your safety net all through this lesson — equivalent expressions must give equal values for every input.
Also watch the case where a variable is outside: is not linear and isn't part of this lesson, but is — distribute to all three terms.
Negative Multipliers and Signs
When the number outside the parentheses is negative, the sign of every term inside flips. That's just the rule that a negative times a positive is negative, and a negative times a negative is positive, applied term by term.
Two things trip people up. First, a bare minus sign in front of parentheses means multiply by — there's an invisible 1 there. Second, when the term inside is already negative, the product becomes positive: gives , not . Students who only flip the first sign end up with , which is a different expression entirely.
Another sign trap involves a subtraction sitting outside the parentheses, like . The being subtracted acts as , so this becomes , which simplifies to . Writing is the classic slip. A reliable habit: before distributing, rewrite the multiplier with its sign attached, so becomes . Now there's nothing to forget.
Check with substitution. At : original is ; our answer . Match.
| Expression | Distribute | Result |
|---|---|---|
| and | ||
| and | ||
| and | ||
| and |
Another sign trap involves a subtraction sitting outside the parentheses, like . The being subtracted acts as , so this becomes , which simplifies to . Writing is the classic slip. A reliable habit: before distributing, rewrite the multiplier with its sign attached, so becomes . Now there's nothing to forget.
Check with substitution. At : original is ; our answer . Match.
Factoring: Pulling Out the Greatest Common Factor
Factoring reverses distribution. Given , you look for the largest number that divides both coefficients — the greatest common factor, or GCF. Here the factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 18 are 1, 2, 3, 6, 9, 18, so the GCF is 6. Divide each term by 6 and write what's left inside parentheses:You can always verify by distributing back: and . If distributing doesn't return the original, something went wrong.
When every term contains the variable, the variable joins the GCF: would first combine to , but factors as . For this lesson most expressions are linear in one variable, so the GCF is usually just a number.
Two frequent mistakes: pulling out a common factor that isn't the greatest one — is true but not fully factored, since still shares a factor of 3 — and forgetting to divide one of the terms, as in . Every term inside must have been divided.
When every term contains the variable, the variable joins the GCF: would first combine to , but factors as . For this lesson most expressions are linear in one variable, so the GCF is usually just a number.
| Expression | GCF | Factored form |
|---|---|---|
| 4 | ||
| 3 | ||
| 1 | already simplest |
Putting It Together: Simplest Form
Most problems in this topic ask for simplest form, meaning no parentheses remain and no two like terms are left uncombined. The order of operations gives you a dependable sequence: distribute first to clear parentheses, then combine like terms.
Work through .
Distribute the : . Distribute the : . Now the whole thing is . Combine the terms: . Combine the constants: . Final answer: .
Notice you cannot combine before distributing — the and the are not like terms with anything, and the terms inside different sets of parentheses aren't next to each other yet.
A question may also run the other direction: simplify, then factor. For instance, expands to , combines to , and factors to . All three forms are equivalent; which one you want depends on what the problem asks.
Where students actually go wrong at this stage: dropping a negative sign in the middle of a long expression, and combining unlike terms such as into . Remember that and are different kinds of terms and the expression is finished. If you're unsure whether your simplified answer is equivalent, substitute a value like into both the original and your answer. Equal values are strong evidence you're right.
Work through .
Distribute the : . Distribute the : . Now the whole thing is . Combine the terms: . Combine the constants: . Final answer: .
Notice you cannot combine before distributing — the and the are not like terms with anything, and the terms inside different sets of parentheses aren't next to each other yet.
A question may also run the other direction: simplify, then factor. For instance, expands to , combines to , and factors to . All three forms are equivalent; which one you want depends on what the problem asks.
Where students actually go wrong at this stage: dropping a negative sign in the middle of a long expression, and combining unlike terms such as into . Remember that and are different kinds of terms and the expression is finished. If you're unsure whether your simplified answer is equivalent, substitute a value like into both the original and your answer. Equal values are strong evidence you're right.
Key terms
- Distributive property.
- The rule : a factor outside parentheses multiplies every term inside.
- Expand.
- To rewrite an expression without parentheses by distributing, for example .
- Factor (verb).
- To rewrite a sum as a product by pulling out a common factor, for example .
- Greatest common factor (GCF).
- The largest factor shared by all terms of an expression; the number (and variable) you pull out when factoring completely.
- Coefficient.
- The number multiplied by a variable in a term; in the coefficient is .
- Like terms.
- Terms with exactly the same variable part, such as and , which can be added or subtracted into one term.
- Equivalent expressions.
- Two expressions that give the same value for every value of the variable, such as and .
- Simplest form.
- An expression with no parentheses left and no like terms left uncombined.
Worked example
Simplify , then factor the result completely.
Step 1: Identify the multiplier with its sign. The is being subtracted, so treat it as . Rewrite: .
Step 2: Distribute the to both terms inside. , and because a negative times a negative is positive. The expression is now .
Step 3: Combine like terms. The terms: . The constants: . Simplest form is .
Step 4: Check by substitution. Use in the original: . Now the answer: . They match.
Step 5: Factor . Factors of 2 are 1 and 2; factors of 22 are 1, 2, 11, 22. The GCF is 2. Divide each term: and . Factored form: .
Step 6: Verify the factoring by distributing back: and . Correct.
Step 2: Distribute the to both terms inside. , and because a negative times a negative is positive. The expression is now .
Step 3: Combine like terms. The terms: . The constants: . Simplest form is .
Step 4: Check by substitution. Use in the original: . Now the answer: . They match.
Step 5: Factor . Factors of 2 are 1 and 2; factors of 22 are 1, 2, 11, 22. The GCF is 2. Divide each term: and . Factored form: .
Step 6: Verify the factoring by distributing back: and . Correct.
Practice questions
Which expression is equivalent to ?
Answer:
Distribute to both terms. . Then , since a negative times a negative is positive. So the result is . The common wrong answer comes from keeping the subtraction sign instead of multiplying the two negatives. Check with : the original is , and .
Simplify completely, then factor your answer.
Answer: , which factors as .
Distribute the : . Distribute the : and , giving . Together: . Combine and , so the simplest form is . The GCF of 8 and 18 is 2 (not 4, since 4 does not divide 18), so factoring gives . Distributing back returns , confirming the work.
Maya says factors as . Explain her mistake and give the correct factored form.
Answer: She only divided the first term by 5. The correct form is .
When you factor out a GCF, every term inside the parentheses must be the original term divided by that GCF. Maya divided by 5 to get but left the 15 unchanged. Distributing her answer gives , which is not the original expression — that check alone reveals the error. Dividing both terms by 5 gives and , so the answer is .
FAQ
- How do I know whether a problem wants me to expand or to factor?
- Look at the form of the expression and the wording. If there are parentheses and the directions say simplify, expand, or write without parentheses, distribute. If the expression is already a sum like and the directions say factor or write as a product, pull out the GCF. Some problems ask for both: simplify first, then factor the result.
- What happens to signs when there's just a minus sign in front of parentheses?
- A bare minus sign means multiply by , so every term inside changes sign. For example, . The invisible 1 is easy to forget, so it helps to actually write the before you distribute.
- Do I have to use the greatest common factor, or is any common factor fine?
- Any common factor produces a true statement, but only the greatest one counts as completely factored. If you factor as , the inside still shares a factor of 3, so you're not finished. The GCF is 6, giving .
- How can I check my answer without the answer key?
- Substitute a number for the variable — 2 or 3 works well, and avoid 0 and 1 since they can hide errors. Evaluate the original expression and your simplified expression. Equivalent expressions must produce the same value. For factoring, just distribute your factored answer and see if you get back the expression you started with.
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The Crimsora tutor teaches Distributive Property & Factoring live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.