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Multiplying over a sum, used forwards to expand and backwards to factor.
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In this lesson you use the distributive law in both directions: forward to expand $3(x + 4)$, and backward to factor $3x + 12$. It is the same law you used to break a factor apart when multiplying, and it will be the law behind expanding and factoring in later grades: one idea that keeps returning in more grown-up clothes.
You can multiply multi-digit numbers by splitting one factor into parts, as in $6 \times 23 = 6 \times 20 + 6 \times 3$. You can find the greatest common factor of two numbers, such as 6 for 18 and 24. You also know that $3x$ means 3 times $x$ and that like terms combine: $2x + 5x = 7x$. This lesson gives the idea behind splitting a factor its proper name, the distributive law, and uses it with letters as well as numbers.
| Term | What it means |
|---|---|
| Distributive law | Multiplying a sum by a number is the same as multiplying each part and adding: $a(b + c) = ab + ac$. |
| Expand | Rewrite a product with parentheses as a sum with none, as in $3(x + 4) = 3x + 12$. |
| Factor (verb) | Rewrite a sum as a product by taking out a common factor, as in $3x + 12 = 3(x + 4)$. |
| Common factor | A number or letter that divides every term of a sum. |
| Greatest common factor (GCF) | The largest number that divides all the given numbers. |
| Area model | A rectangle split into parts, used to show a product as a sum of smaller areas. |
Suppose 4 friends each buy a sandwich for 6 dollars and a juice for 2 dollars. You can find the total in two ways. Add first: each friend spends $6 + 2 = 8$ dollars, so the total is $4 \times 8 = 32$. Or multiply first: the sandwiches cost $4 \times 6 = 24$ and the juices $4 \times 2 = 8$, so the total is $24 + 8 = 32$. The same money, counted two ways:
$$4(6 + 2) = 4 \times 6 + 4 \times 2$$
That is the distributive law. A number multiplying a sum multiplies every part of the sum. In letters, for any numbers $a$, $b$ and $c$:
$$a(b + c) = ab + ac \qquad a(b - c) = ab - ac$$
Because it is true for every number, it works when one part is a variable. $4(x + 2)$ means 4 groups of $x + 2$, which is 4 $x$'s and 4 twos: $4x + 8$. We cannot add $x$ and 2 first, since we do not know $x$, but the distributive law lets us get rid of the parentheses anyway.
The law works in both directions. Going left to right, $4(x + 2) \to 4x + 8$, is called expanding. Going right to left, $4x + 8 \to 4(x + 2)$, is called factoring. Both give equivalent expressions.
Another way: picture
Draw a rectangle 4 units tall and $x + 2$ units long. Cut it where the $x$ part ends. The left piece is 4 by $x$, with area $4x$; the right piece is 4 by 2, with area 8. The whole rectangle has area $4(x + 2)$, and the pieces have $4x + 8$. It is the same rectangle, so the areas are equal.
Another way: story
A teacher hands out a pencil and 3 stickers to each of the 25 students in a class. She can count 25 bundles of $1 + 3$ things, which is $25 \times 4 = 100$, or count 25 pencils and 75 stickers, which is also 100. Either count tells her how many things she needs.
To expand, multiply the number outside the parentheses by each term inside, one at a time, and keep the plus or minus sign between them.
In the middle example, $2 \times 4k$ is $8k$: multiply the numbers and keep the letter, because 2 groups of 4 $k$'s is 8 $k$'s.
The most common mistake is to multiply only the first term: writing $5(y + 3) = 5y + 3$. Think of the friends buying lunch. If you multiply the sandwiches by the number of friends and forget the juices, only one friend gets a drink. Draw an arrow from the outside number to each term inside if it helps you remember.
The multiplier can also be written on the right: $(y + 3)5$ is the same as $5(y + 3)$, because the order of a multiplication does not matter.
Factoring undoes expanding. Look at $6x + 15$. Both terms are multiples of 3: $6x = 3 \times 2x$ and $15 = 3 \times 5$. Read the distributive law from right to left and the 3 comes out in front:
$$6x + 15 = 3(2x + 5)$$
To factor completely, take out the greatest common factor. For $12y + 18$, the number 2 divides both, giving $2(6y + 9)$. That is equivalent, but 6 and 9 still share a 3, so it is not finished. The GCF of 12 and 18 is 6, and $12y + 18 = 6(2y + 3)$. Now 2 and 3 share nothing except 1, and the job is done.
Always check a factoring by expanding it again. $6(2y + 3) = 12y + 18$, which is where you started. If expanding does not bring back the original, something was divided wrongly.
Factoring can also pull out a letter. In $7n + 4n$, both terms have the factor $n$, so $7n + 4n = (7 + 4)n = 11n$. This is really why combining like terms works: it is the distributive law in disguise.
The distributive law is the secret behind many mental-math tricks. To find $8 \times 104$, split 104 into $100 + 4$: $8 \times 100 + 8 \times 4 = 800 + 32 = 832$. To find $6 \times 49$, think of 49 as $50 - 1$: $6 \times 50 - 6 \times 1 = 300 - 6 = 294$.
The trick is to split a number into a friendly part and a small leftover. Numbers close to 10, 50, 100 or 1,000 are good candidates. Subtracting works as well as adding, as long as the small part is multiplied too: $6 \times 49$ is not $300 - 1$.
Money is a perfect place to use it. Prices like 2.99 dollars are just 1 cent under 3 dollars, so 7 of them cost $7 \times 3 - 7 \times 0.01 = 21 - 0.07 = 20.93$ dollars.
Multiplication by a whole number is repeated addition, and that is the whole reason the law works. $3(x + 4)$ means three copies of $x + 4$ added together:
$$(x + 4) + (x + 4) + (x + 4)$$
Because a sum can be reordered and regrouped, the three $x$'s can be gathered together and the three 4s can be gathered together: $x + x + x + 4 + 4 + 4$. That is $3x + 12$. Nothing was added or lost; the same six pieces were only sorted into two piles.
The area model tells the same story for any numbers, even fractions and decimals. A rectangle cut into two pieces has the same total area as the two pieces put together, whatever the lengths are. So the law is not a trick that happens to work for the examples you try. It holds for every value of the variable, which is exactly what makes the two sides equivalent expressions.
To expand $a(b + c)$:
To factor a sum such as $ab + ac$:
How to check. Expanding and factoring are opposites, so each checks the other: expand your factored answer and you must get the original back. You can also substitute a value, such as $x = 3$, into both forms; they must give the same number. A mismatch usually means a term inside the parentheses was not multiplied, or a minus sign got lost.
In many American restaurants people leave a tip of about 15 to 20 percent. The distributive law makes a 15% tip easy to find in your head, because $15\% = 10\% + 5\%$:
$$0.15 \times 48 = (0.10 + 0.05) \times 48 = 0.10 \times 48 + 0.05 \times 48$$
Ten percent of a 48 dollar bill is 4.80 dollars: just move the decimal point one place left. Five percent is half of that, 2.40 dollars. So the tip is $4.80 + 2.40 = 7.20$ dollars.
A 20% tip is even quicker, $2 \times 10\%$: $2 \times 4.80 = 9.60$ dollars. And an 18% tip can be found as $20\% - 2\%$: $9.60 - 0.96 = 8.64$ dollars. Each time, an awkward percent is split into friendly pieces, each piece is multiplied by the bill, and the results are added or subtracted. That is exactly $a(b + c) = ab + ac$.
Stores often set prices just under a whole number of dollars, such as 4.99 dollars. When you buy several, the distributive law finds the total fast. Six items at 4.99 dollars each cost
$$6 \times 4.99 = 6 \times (5 - 0.01) = 30 - 0.06 = 29.94 \text{ dollars}.$$
Or suppose a class orders 24 T-shirts at 12.99 dollars each. Split the price into $13 - 0.01$: $24 \times 13 = 312$ and $24 \times 0.01 = 0.24$, so the order costs $312 - 0.24 = 311.76$ dollars. A cashier who knows the trick can check a register total in seconds, and a shopper can tell whether 20 dollars will cover four items at 4.99 dollars: $4 \times 5 = 20$, less 4 cents, so yes, with 4 cents to spare.
Multiplying only the first term. $3(x + 5)$ is $3x + 15$, not $3x + 5$. The 3 reaches every term.
Losing the minus sign. $4(y - 2)$ is $4y - 8$, not $4y + 8$.
Adding the outside number. $2(x + 6)$ is not $x + 8$ or $2 + x + 6$. The number against the parentheses multiplies.
Stopping before the greatest common factor. $2(9a + 6)$ equals $18a + 12$, but 9 and 6 still share 3. The complete factoring is $6(3a + 2)$.
Dividing only one term when factoring. $10x + 15 = 5(2x + 15)$ is wrong: expand it and you get $10x + 75$. Every term is divided by the common factor.
Mental math with a subtraction. $5 \times 99$ is $500 - 5 = 495$, not $500 - 1$. The small part is multiplied too.
Expand $4(x + 6)$. Picture a rectangle 4 wide and $x + 6$ long.
$\text{area} = 4(x + 6)$
Width times length gives the area of a rectangle.
Find the area of the part that is $x$ long.
$4 \times x = 4x$
The first piece is 4 by $x$.
Find the area of the part that is 6 long.
$4 \times 6 = 24$
The second piece is 4 by 6.
Add the two areas.
$4(x + 6) = 4x + 24$
The two pieces make up the whole rectangle.
Check both forms with $x = 5$.
$4(5 + 6) = 44, \qquad 4(5) + 24 = 44$
Equivalent expressions give the same value.
Expand $3(5y - 2)$. Identify the terms inside.
$5y \quad\text{and}\quad -2$
The 3 must multiply each one.
Multiply the first term.
$3 \times 5y = 15y$
3 groups of 5 $y$'s make 15 $y$'s.
Multiply the second term.
$3 \times 2 = 6$
Each of the 3 groups is 2 short, so 6 is taken away altogether.
Write the expansion with the minus sign.
$3(5y - 2) = 15y - 6$
The subtraction inside stays a subtraction outside.
Check the original with $y = 2$.
$3(10 - 2) = 3 \times 8 = 24$
Working the parentheses first gives the value of the original.
Check the expansion with $y = 2$.
$15(2) - 6 = 30 - 6 = 24$
Both give 24, so no term was missed and no sign was lost.
Factor $24x + 18y$ completely. List the factors of 18, the smaller coefficient.
$1, 2, 3, 6, 9, 18$
Any common factor must be a factor of both numbers, so start with the smaller one.
Find the largest of those that also divides 24.
$24 \div 9 \text{ and } 24 \div 18 \text{ leave remainders; } 24 \div 6 = 4$
Testing from the largest down finds the greatest common factor, 6.
Divide the first term by 6.
$24x \div 6 = 4x$
This is what is left of the first term inside the parentheses.
Divide the second term by 6.
$18y \div 6 = 3y$
The letter stays with its term.
Write the factored form.
$24x + 18y = 6(4x + 3y)$
The distributive law read backward puts the common factor outside.
Check that nothing more can come out.
$\text{GCF}(4, 3) = 1$
If 4 and 3 shared a factor, a bigger number could have come out.
Check by expanding.
$6 \times 4x + 6 \times 3y = 24x + 18y$
Expanding brings back the original, so the factoring is right.
Multiply the first term inside.
$8 \times 2 = 16$
The 8 reaches the 2.
Multiply the second term inside.
$8 \times k = 8k$
The 8 reaches the $k$ as well.
Write the expansion.
Use the distributive law to expand $8(5y - 6)$.
Answer:
A rectangle is 9 units wide and $8x + 7$ units long. Complete the solution to expand $9(8x + 7)$ using its area, then check it at $x = 6$.
Split the length into its two parts.
$\text{length} = 8x + 7$
Cutting the rectangle there makes two smaller rectangles, both 9 units wide.
Find the coefficient of $x$ in the first part's area.
$9 \times 8 =$ coef
Area is width times length, and the first part is $8x$ long.
Find the second part's area.
$9 \times 7 =$ const
The second part is 7 long and has the same width.
Add the two areas.
$\text{whole area} = \text{first part} + \text{second part}$
The two parts fill the whole rectangle with no overlap, which is why the distributive law works.
Check by working out the area at $x = 6$ from the unexpanded form.
$9 \times (8 \times 6 + 7) =$ check
Equivalent expressions must give the same area; the expanded form gives this number too.
Work out $4 \times 197$ in your head by writing 197 as $200 - 3$.
$4 \times 197 =$ answer
Factor $27a + 72$ by taking out the greatest common factor.
27a + 72 = g(ma + n)
Expand and simplify $6(x + 9) + 9x$.
Answer:
A group of 8 friends go bowling. Each one pays $t$ dollars for a game and 10 dollars for shoe rental. The total cost is $8(t + 10)$ dollars. Write the total cost without parentheses.
Answer:
One batch of pancakes uses 22 ounces of flour and 14 ounces of milk. Write, as simply as you can, the total number of ounces of flour and milk for $n$ batches.
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A group of 3 friends go bowling. Each one pays $t$ dollars for a game and 12 dollars for shoe rental. The total cost is $3(t + 12)$ dollars. Write the total cost without parentheses.
Answer:
You can expand and factor using the distributive law. Without looking: factor $6x + 15$, and say what you took out and how you found it.
16. Your turn: expand $8(2 + k)$, step 3
$8(2 + k) = 16 + 8k$
The two products are added, as the terms inside were.