Distributive Property

📦 The Distributive Property

Part 1 of 5 — The Core Idea

Topics in This Part

🔑 Key Concept: The distributive property says a number multiplied by a sum equals the sum of the products: $a(b + c) = ab + ac$ You "hand out" the multiplier to every term inside the parentheses.

What "Distribute" Means

When a number sits right next to a parenthesis, it multiplies everything inside:

$3(x + 5) = 3 \cdot x + 3 \cdot 5 = 3x + 15$

The $3$ gets "distributed" to the $x$ and to the $5$ — not just the first term.

A Mental Picture

Think of $3(x + 5)$ as three copies of the group $(x + 5)$:

$(x+5) + (x+5) + (x+5) = 3x + 15$

Three $x$'s and three $5$'s — exactly what distributing gives you.

⚠️ Common Mistake: $3(x + 5) \ne 3x + 5$. The $3$ must reach the $5$ too. Forgetting the second term is the #1 distribution error.

The Area Model — Why It's Always True

A rectangle's area is length $\times$ width. Split the width into two pieces and the total area is the sum of the two smaller rectangles:

The whole rectangle has height $a$ and total width $b + c$, so its area is $a(b+c)$. But it's also made of the two pieces $ab$ and $ac$. Therefore:

$a(b + c) = ab + ac$

Numeric Check

Take $a = 4$, $b = 10$, $c = 3$:

$4(10 + 3) = 4 \cdot 13 = 52 \qquad\text{and}\qquad 4\cdot 10 + 4\cdot 3 = 40 + 12 = 52 \;✓$

💡 This is also the mental-math trick for multiplication: $4 \times 13 = 4(10+3) = 40 + 12 = 52$.

Concept Check 🎯

Writing It Out Step by Step

A reliable habit: write the two products before simplifying, so nothing gets skipped.

$8(x + 6) \;=\; \underbrace{8 \cdot x}_{8x} \;+\; \underbrace{8 \cdot 6}_{48} \;=\; 8x + 48$

Now try a few on your own — focus on multiplying the outside number by the constant each time.

Distribute It 🧮

Multiply out each expression. Enter the constant (the number term) for each.

1) $2(x + 7) = 2x + \,?$ 2) $4(a + 3) = 4a + \,?$ 3) $9(y + 6) = 9y + \,?$

Both Terms Have a Variable? Same Rule

The multiplier still reaches every term, even when the inside has more than a constant:

$4(2x + 3) = 4 \cdot 2x + 4 \cdot 3 = 8x + 12$

Notice $4 \cdot 2x = 8x$ — multiply the coefficients ($4 \cdot 2 = 8$) and keep the variable.

Example: $7(3m + 1) = 21m + 7$.

You now have the one rule the whole topic is built on: reach every term. In Part 2 we add the twist that trips up the most students — negative signs.

Match the Distribution 🔽

Choose the correct expanded form for each.

📦 The Distributive Property

Part 2 of 5 — Negatives & the Minus Sign

🔑 The Idea: A negative multiplier flips the sign of every term it touches. And a subtraction inside the parentheses is really "plus a negative," so it gets distributed too.

A Negative on the Outside

When the multiplier is negative, apply the sign rules to each product:

$-2(x + 5) = (-2)(x) + (-2)(5) = -2x - 10$

$-3(4a + 7) = -12a - 21$

Both terms become negative because a negative times a positive is negative.

Subtraction Inside

Treat $(x - 4)$ as $(x + (-4))$:

$5(x - 4) = 5x - 20$

$6(2y - 3) = 12y - 18$

⚠️ Watch the sign of the second term. In $5(x-4)$, the $-4$ stays negative: $5 \cdot (-4) = -20$.

Negative Outside and Subtraction Inside

This is where signs matter most. A negative times a negative is positive:

$-4(x - 3) = (-4)(x) + (-4)(-3) = -4x + 12$

The $-4 \cdot -3 = +12$ — two negatives make a positive.

Worked Example: $-2(3x - 5)$

$-2(3x - 5) = (-2)(3x) + (-2)(-5) = -6x + 10$

💡 Sign shortcut: When the outside number is negative, every inside sign flips. $+$ becomes $-$, and $-$ becomes $+$.

Pick the Right Sign 🔽

Distribute each expression and choose the correct result.

Two Sign Traps to Avoid

Most sign mistakes come from one of these two slips:

Read the outside sign and the inside sign together for each term. Let's test that on a fresh set.

Concept Check 🎯

Now Without the Answer Choices

Multiple choice lets you check against options. The real skill is producing the right sign on your own. In the drill below, compute each constant term yourself — outside number times the inside constant, keeping track of the sign.

Reminder: $(\text{negative}) \times (\text{positive}) = \text{negative}$, and $(\text{negative}) \times (\text{negative}) = \text{positive}$.

Sign Drill 🧮

Distribute and enter the constant term, with its sign (e.g. -12).

1) $-4(x + 5) = -4x + \,?$ 2) $3(2a - 9) = 6a + \,?$ 3) $-7(m - 2) = -7m + \,?$

📦 The Distributive Property

Part 3 of 5 — Combine, Then Factor

🔑 Two directions: Distributing expands a product into a sum. After distributing you usually combine like terms. And running the rule backward lets you factor — pull a common factor back out front.

Distribute, Then Combine Like Terms

Like terms have the same variable part. After distributing, add or subtract them.

Worked Example: $3(x + 4) + 2x$

$3(x + 4) + 2x = 3x + 12 + 2x = (3x + 2x) + 12 = 5x + 12$

Worked Example: $5(2a + 1) - 3a$

$5(2a + 1) - 3a = 10a + 5 - 3a = 7a + 5$

💡 Only combine terms that match: $x$'s with $x$'s, plain numbers with plain numbers. You cannot combine $5x$ and $12$ — they're unlike.

Two Sets of Parentheses

Distribute each group first, then combine:

Worked Example: $2(x + 3) + 4(x + 1)$

$2(x+3) + 4(x+1) = (2x + 6) + (4x + 4) = 6x + 10$

Worked Example: $6(a + 2) - 3(a + 1)$

Watch the $-3$ — it distributes as a negative:

$6(a+2) - 3(a+1) = (6a + 12) + (-3a - 3) = 3a + 9$

⚠️ The minus in front of $3(a+1)$ makes it $-3a - 3$. Distribute the negative, then combine: $6a - 3a = 3a$ and $12 - 3 = 9$.

Concept Check 🎯

Factoring — Distribution in Reverse

To factor, pull out the greatest common factor (GCF) of all terms and write it in front:

$6x + 9 = 3(2x + 3)$

Here $3$ divides both $6x$ and $9$: $\;6x \div 3 = 2x$ and $9 \div 3 = 3$.

How to Factor Out a GCF

1. Find the largest number that divides every coefficient.
2. Write it outside the parentheses.
3. Inside, write what's left after dividing each term.

✅ Check by re-distributing: $5(2a + 3) = 10a + 15$ ✓. Factoring and distributing undo each other.

Factor It Out 🔽

Choose the fully factored form (largest possible GCF).

Find the GCF First

The hardest part of factoring is spotting the largest common factor. Always ask: "What is the biggest number that divides every coefficient?"

Pull out the greatest one — then the leftover inside has no remaining common factor. That's how you know you factored completely.

GCF Drill 🧮

Factor out the greatest common factor. Enter the number that goes outside the parentheses.

1) $20x + 30 = \,?\,(2x + 3)$ 2) $24a - 16 = \,?\,(3a - 2)$ 3) $18m + 27 = \,?\,(2m + 3)$

📦 The Distributive Property

Part 4 of 5 — Real-World Uses & Bigger Expressions

🔑 Where it shows up: Mental math, shopping totals, perimeter and area, and setting up equations all lean on distribution. Let's use it on real problems.

Mental Math with Distribution

Break a hard product into easy pieces:

$7 \times 98 = 7(100 - 2) = 700 - 14 = 686$

$6 \times 53 = 6(50 + 3) = 300 + 18 = 318$

Shopping Example

You buy $4$ notebooks, each costing $3 for the notebook plus $1 tax. The total is:

$4(3 + 1) = 4 \cdot 3 + 4 \cdot 1 = 12 + 4 = 16$

giving $16. Either pay $4 per notebook four times, or buy $12 of notebooks and $4 of tax — same total. That's the distributive property in your wallet.

Mental Math Drill 🧮

Use the distributive property to compute each (no calculator needed).

1) $5 \times 99 = 5(100 - 1) = \,?$ 2) $8 \times 21 = 8(20 + 1) = \,?$ 3) $3 \times 104 = 3(100 + 4) = \,?$

Geometry: Perimeter & Area

A rectangle is $x + 4$ long and $3$ wide.

Area $=$ length $\times$ width:

$3(x + 4) = 3x + 12$

Perimeter $= 2(\text{length} + \text{width})$:

$2\big((x + 4) + 3\big) = 2(x + 7) = 2x + 14$

💡 Distribution turns a "grouped" geometry formula into a clean expression you can evaluate for any $x$. If $x = 6$: area $= 3(6)+12 = 30$ and perimeter $= 2(6)+14 = 26$.

Application Check 🎯

Three Terms Inside

The rule reaches every term, no matter how many:

$2(x + 3y + 5) = 2x + 6y + 10$

$-3(2a - b + 4) = -6a + 3b - 12$

In the second one: $-3 \cdot 2a = -6a$, $\;-3 \cdot (-b) = +3b$, $\;-3 \cdot 4 = -12$.

⚠️ With three or more terms, students often stop after two. Touch each term, every time.

Multi-Term Drill 🧮

Distribute fully. Enter each term's coefficient/constant with sign, in order.

1) $4(a + 2b + 3)$: the $a$-coefficient is ?, the $b$-coefficient is ?, the constant is ?

📦 The Distributive Property

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) distribute a multiplier to every term, (2) handle negatives and subtraction, (3) combine like terms, (4) factor out a GCF, and (5) apply it to real problems. Time to put it all together.

Quick Reference

⚠️ Three things to never forget: reach every term, watch the sign of a negative multiplier, and only combine like terms.

Mixed Practice 🎯

A Reliable Order of Operations

When an expression mixes distribution and combining, work in this order:

1. Distribute every multiplier across its parentheses (watch the signs).
2. Combine like terms — $x$'s together, constants together.
3. Double-check by re-distributing a factored answer, or by plugging in a number.

💡 Plug-in check: To test $3(2x+1)+4x = 10x+3$, try $x=1$: left $= 3(3)+4 = 13$, right $= 10+3 = 13$ ✓.

Mixed Drill 🧮

1) Simplify $3(2x + 1) + 4x$ and enter the $x$-coefficient: ? 2) ... and enter its constant: ? 3) Compute $7 \times 96$ using distribution: ?

You're Ready

You've expanded, handled negatives, combined like terms, factored out a GCF, and used distribution on real problems. One final set ties it all together — take your time and watch every sign.

Exit Quiz ✅

Answer all three to finish the lesson.