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The distributive law, and the sign that gets lost in front of a bracket.
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In this lesson you expand a bracket such as $3(2x - 5)$ by multiplying everything inside it by what is outside. The case to slow down for is a minus sign in front: $-2(x - 4)$ changes both signs inside, and forgetting the second is the single most common error in school algebra. Expanding is the distributive property, the same rule that let you break a factor apart in grade 3.
You have used the distributive property with plain numbers for years, maybe without its name. To work out $7 \times 24$ in your head, you can do $7 \times 20 = 140$ and $7 \times 4 = 28$, then add: $168$. You can multiply positive and negative numbers: $-4 \times 5 = -20$ and $-4 \times (-5) = 20$. And you can combine like terms, such as $5x + 2x = 7x$ and $9 - 3 = 6$. This lesson puts those three skills together to remove brackets from algebra expressions. You will need it in the next lesson, where many equations have a bracket that must come off before you can solve them.
| Term | What it means |
|---|---|
| Expand | Rewrite an expression so it has no brackets, by multiplying out. |
| Distributive property | The rule $a(b + c) = ab + ac$: a factor in front of a bracket multiplies every term inside. |
| Term | One part of an expression, with its sign, separated from the others by $+$ or $-$. In $3x - 8$ the terms are $3x$ and $-8$. |
| Coefficient | The number multiplying a letter. In $-6x$ the coefficient is $-6$. |
| Like terms | Terms with the same letter part, such as $4x$ and $-x$, or two plain numbers. They can be combined. |
| Equivalent expressions | Expressions that give the same value for every value of the letter, such as $2(x + 5)$ and $2x + 10$. |
A bracket with a number in front, like $4(x + 3)$, means four copies of $x + 3$:
$$(x + 3) + (x + 3) + (x + 3) + (x + 3) = 4x + 12.$$
Four copies of $x$ make $4x$, and four copies of $3$ make $12$. The shortcut is the distributive property: the factor in front multiplies every term inside the bracket, and the products are added.
$$a(b + c) = ab + ac$$
Expanding gives an equivalent expression. The two forms, $4(x + 3)$ and $4x + 12$, look different but give the same number for every value of $x$. Try $x = 5$: $4(5 + 3) = 32$ and $4 \times 5 + 12 = 32$. That is why you are allowed to swap one for the other in the middle of a problem.
Signs travel with their terms. In $6(x - 2)$ the terms inside are $x$ and $-2$, so the products are $6x$ and $-12$. And a minus sign in front of the bracket is part of the factor: in $-5(x - 2)$ the factor is $-5$, and it multiplies both $x$ and $-2$.
Another way: picture
Draw a rectangle $4$ units tall and $x + 3$ units wide. Split the width into a piece $x$ wide and a piece $3$ wide. The two smaller rectangles have areas $4x$ and $12$. The big rectangle's area is $4(x + 3)$, so $4(x + 3) = 4x + 12$.
Another way: numbers
Mental math uses the same law: $8 \times 99 = 8(100 - 1) = 800 - 8 = 792$.
Start with the plain case, where the factor in front is a positive number. Take $5(2x + 7)$. The terms inside are $2x$ and $7$.
So $5(2x + 7) = 10x + 35$. If there is a minus sign between the terms, it belongs to the second term: $3(4x - 9)$ has terms $4x$ and $-9$, so it expands to $12x - 27$.
The factor can be on the right, too: $(x + 6) \times 2 = 2x + 12$. And the letter can be on the outside: $x(x + 4) = x^2 + 4x$, because $x \times x$ is $x^2$. In every case the rule is the same. Draw an arrow from the factor to each term inside, one arrow per term, and write one product for each arrow. If there are three terms inside, there are three arrows: $2(x + 3y - 1) = 2x + 6y - 2$.
This is where most mistakes happen. When a bracket has a minus sign in front, the minus is part of the factor, and it changes the sign of every term inside.
Take $-3(x - 5)$. The factor is $-3$. The terms inside are $x$ and $-5$.
So $-3(x - 5) = -3x + 15$. The common wrong answer is $-3x - 15$, which changes the first sign and forgets the second.
A bracket with only a minus in front, like $-(x - 8)$, has a hidden factor of $-1$. It expands to $-x + 8$: every sign inside flips. The same thing happens when you subtract a whole bracket in a longer expression: $10 - (x + 4)$ is $10 - x - 4$, which is $6 - x$. Taking away the whole of $x + 4$ means taking away the $x$ and taking away the $4$.
Often an expression has a bracket and other terms too, or two brackets. Then there are two stages: first expand every bracket, then collect like terms.
Take $4(x + 2) + 3x - 5$. Expanding gives $4x + 8 + 3x - 5$. The $3x$ and $-5$ were outside the bracket, so they are not multiplied by $4$. Now collect: $4x + 3x = 7x$ and $8 - 5 = 3$. The answer is $7x + 3$.
With two brackets, expand each one with its own factor. For $2(3x + 1) + 4(x - 2)$: the first gives $6x + 2$, the second gives $4x - 8$, and together $10x - 6$.
Why expand before collecting? Because the bracket groups its terms. In $4(x + 2) + 3x$, you cannot add the $3x$ to the $x$ inside the bracket: that $x$ is really $4x$ until the bracket comes off.
Every expansion can be run in reverse. Running it backward is called factoring, and it is a good way to see that the two forms really are the same expression.
Take $12x + 18$. Look for a number that divides both terms. Both $12$ and $18$ are multiples of $6$, so write each term as $6$ times something: $12x = 6 \times 2x$ and $18 = 6 \times 3$. Now the $6$ can go in front of a bracket: $12x + 18 = 6(2x + 3)$. Expanding $6(2x + 3)$ gives back $12x + 18$, which checks the factoring.
Factoring and expanding answer different questions. The expanded form $12x + 18$ shows the rate and the starting amount, the way a linear model does. The factored form $6(2x + 3)$ shows that the value is always a multiple of $6$, whatever whole number $x$ is. Being able to move in both directions lets you pick whichever form makes the next step easier. In this course you will mostly expand, because solving an equation goes more smoothly once every bracket is gone.
Why each move is allowed. Step 3 is the distributive property, which is true for every number, so the new expression is equivalent to the old one. Step 5 only adds terms that count the same thing, like adding $4$ apples to $3$ apples.
How to check. Pick an easy value, such as $x = 1$ or $x = 2$, and put it into both the original expression and your answer. They must give the same number. For $-3(x - 5) = -3x + 15$ with $x = 2$: $-3(2 - 5) = -3 \times (-3) = 9$, and $-3 \times 2 + 15 = 9$. They match. If they do not match, look first at the signs of the terms from a negative factor. Avoid checking with $x = 0$ alone, since it hides mistakes in the $x$ term.
Store prices often end in $99$ cents, and the distributive property turns them into easy mental math. Six notebooks at $\$2.99$ each cost $6(3 - 0.01)$ dollars. Expanding gives $6 \times 3 - 6 \times 0.01 = 18 - 0.06 = \$17.94$. Four movie tickets at $\$12.50$ each are $4(12 + 0.5) = 48 + 2 = \$50$. A cashier who knows the property can check a total in seconds: round each price to the nearest dollar, multiply, then fix the small difference, exactly as the expanded form says.
Most US states charge a sales tax, a percent added to the price. With a tax rate of $7\%$, an item priced $p$ dollars costs $p + 0.07p$ in all. The distributive property in reverse (factoring) turns that into $p(1 + 0.07) = 1.07p$. So you can find the total in one step: a $\$40$ jacket costs $1.07 \times 40 = \$42.80$. For a whole cart, the property works the other way: $1.07(25 + 15 + 8) = 1.07 \times 25 + 1.07 \times 15 + 1.07 \times 8$. Taxing each item and adding gives the same total as adding first and taxing once, $1.07 \times 48 = \$51.36$.
A community garden plot is $x$ feet wide and $x + 6$ feet long. Its perimeter is $2(x + x + 6)$ feet: twice the width plus the length. Expanding gives $2x + 2x + 12 = 4x + 12$. That expanded form tells the gardener something useful: every extra foot of width adds $4$ feet of fence, and there are always $12$ extra feet because the plot is longer than it is wide. For a plot $10$ feet wide, the fence is $4 \times 10 + 12 = 52$ feet. At $\$3$ a foot, that is $\$156$, and the cost is $3(4x + 12) = 12x + 36$ dollars in general.
Multiplying only the first term. $6(x + 4)$ is $6x + 24$, not $6x + 4$. The factor reaches every term.
Forgetting the second sign change. $-2(x - 7)$ is $-2x + 14$, not $-2x - 14$. Negative times negative is positive.
Multiplying terms outside the bracket. In $3(x + 1) + 5$, the $5$ is not multiplied by $3$.
Adding instead of multiplying. $4(x + 2)$ is not $4 + x + 2$. The factor multiplies.
Collecting before expanding. In $2(x + 3) + x$, you cannot add $x$ to the $x$ inside first.
Expand $6(2x + 5)$. Name the factor and the terms inside.
$\text{factor } 6; \quad \text{terms } 2x \text{ and } 5$
Knowing every term inside tells you how many products to write.
Multiply the factor by the $x$ term.
$6 \times 2x = 12x$
Multiply the numbers; the $x$ comes along.
Multiply the factor by the number term.
$6 \times 5 = 30$
The factor reaches the second term too.
Write the expanded expression.
$6(2x + 5) = 12x + 30$
One product for each term inside, joined by the sign between them.
Check with $x = 3$.
$6(6 + 5) = 66; \quad 36 + 30 = 66$
Equivalent expressions give the same value for any $x$.
Expand $-3(5 - 4x)$. Name the factor, with its sign.
$\text{factor } -3$
The minus sign in front of the bracket belongs to the factor.
List the terms inside, with their signs.
$5 \text{ and } -4x$
The minus between them belongs to the $4x$.
Multiply the factor by the first term.
$-3 \times 5 = -15$
Negative times positive is negative.
Multiply the factor by the second term.
$-3 \times (-4x) = 12x$
Negative times negative is positive.
Write the products together.
$-15 + 12x$
Each product keeps the sign it came out with.
Put the $x$ term first.
$12x - 15$
Adding works in any order, and the $x$ term first is the usual form.
Check with $x = 2$.
$-3(5 - 8) = 9; \quad 24 - 15 = 9$
Both forms give $9$, so the signs are right.
Simplify $5(2x + 3) - 2(4x - 1) + 6$. Split it into its parts.
$5(2x + 3), \quad -2(4x - 1), \quad +6$
Each bracket has its own factor; the $6$ is outside both.
Multiply the first $x$ term.
$5 \times 2x = 10x$
The factor $5$ reaches the $2x$.
Multiply the first number term.
$5 \times 3 = 15$
And it reaches the $3$.
Multiply the second $x$ term by $-2$.
$-2 \times 4x = -8x$
The minus in front belongs to the factor $-2$.
Multiply the second number term by $-2$.
$-2 \times (-1) = 2$
Negative times negative is positive.
Write every term in a row.
$10x + 15 - 8x + 2 + 6$
No brackets are left. The $6$ was never multiplied.
Collect the $x$ terms.
$10x - 8x = 2x$
Like terms combine by their coefficients.
Collect the numbers.
$15 + 2 + 6 = 23$
All three numbers are like terms.
Write the result.
$2x + 23$
One $x$ term and one number: fully simplified.
Check with $x = 1$.
$5(5) - 2(3) + 6 = 25; \quad 2 + 23 = 25$
The original and the answer agree.
Multiply the $x$ term inside by $3$.
$3 \times 4x = 12x$
The factor multiplies each term inside.
Multiply the number inside by $3$.
$3 \times 1 = 3$
The $1$ is inside the bracket, so it is multiplied too.
Write every term.
Collect the $x$ terms.
A student expands $5(x - 4)$ as $5x - 4$. What did they miss?
Complete the worked solution to expand and simplify $5(x + 4) + 2(x + 3)$.
Multiply the number inside the first bracket by $5$.
$5 \times 4 =$ m
The factor outside multiplies every term inside, numbers included.
Multiply the number inside the second bracket by $2$.
$2 \times 3 =$ n
The second bracket has its own factor.
Collect the $x$ terms.
$5x + 2x =$ k$x$
Like terms add their coefficients.
Add the two numbers found in the first two steps.
number term $=$ t
The numbers are like terms too, so they combine into one.
Expand the bracket: $5(5x - 4)$.
Answer:
Expand: $-6(3x + 3)$.
Answer:
Expand and simplify: $4(5x + 7) - 7$.
Answer:
Expand and simplify: $8(x + 5) - 2(x - 7)$.
Answer:
A school orders $5$ identical art kits. Each kit holds $2$ markers costing $x$ dollars each and one carrying case costing $4$ dollars. Write the total cost of the order, in dollars, as an expression with no brackets.
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Expand: $-5(3x - 7)$.
Answer:
You can expand a bracket, including one with a minus in front. Without looking: expand $-2(x - 4)$, and say what happened to each sign.
17. Your turn: expand and simplify $3(4x + 1) + 2x$., step 3
$12x + 3 + 2x$
The $2x$ was outside the bracket, so it is not multiplied.
17. Your turn: expand and simplify $3(4x + 1) + 2x$., step 4
$14x + 3$
$12x + 2x = 14x$.