3.02 The distributive property with algebraic terms
Remember that the order of operations calls for us to evaluate expressions inside grouping symbols (such as parenthesis) first. However, sometimes the terms inside the grouping symbol cannot be combined. Is there a different way that we might rewrite the expression?
Exploration
Let's use the applet below to rewrite a few expressions with parenthesis. Set the sliders to get the expression in the first column of the table. Then, count the number of $+x$+x tiles and $+1$+1 tiles to complete the rest:
- What patterns exist between the number of $+x$+xs, $+1$+1s and the expression in the last column?
- If someone says that $2\left(3x+3\right)=6x+3$2(3x+3)=6x+3, should we agree or disagree? Why?
| Expression | Number of $+x$+x Tiles | Number of $+1$+1 Tiles | $=$= |
|---|---|---|---|
| $1\left(4x+3\right)$1(4x+3) | $4$4 | $3$3 | $4x+3$4x+3 |
| $2\left(4x+3\right)$2(4x+3) | $8$8 | $6$6 | $8x+6$8x+6 |
| $3\left(4x+3\right)$3(4x+3) | |||
| $4\left(4x+3\right)$4(4x+3) | |||
| $1\left(2x+1\right)$1(2x+1) | $2$2 | $1$1 | $2x+1$2x+1 |
| $2\left(2x+1\right)$2(2x+1) | $4$4 | $4$4 | $4x+4$4x+4 |
| $3\left(2x+1\right)$3(2x+1) | |||
| $4\left(2x+1\right)$4(2x+1) |
If we complete the table using the applet above, we can see that we multiply everything inside the parenthesis by the first number.
Therefore, the expression $2\left(3x+3\right)$2(3x+3) should be equal to $2$2 groups of $3x+3$3x+3, or $2$2 groups of $3x$3x and $2$2 groups of $3$3. This means it is equal to $6x+6$6x+6!
Here's another way we might look at rewriting the expression using the associative and commutative properties of addition.
| $2\left(3x+3\right)$2(3x+3) | $=$= | $($($3x+3$3x+3$)$)$+$+$($($3x+3$3x+3$)$) |
Two groups of $3x+3$3x+3 |
| $=$= | $3x+3+3x+3$3x+3+3x+3 |
Associative property (addition can be grouped differently) |
|
| $=$= | $3x+3x+3+3$3x+3x+3+3 |
Commutative property (we can add in any order) |
|
| $=$= | $6x+6$6x+6 |
Simplify |
The property that we have just demonstrated is called the distributive property of real numbers. We can summarize it in the box below:
For all numbers $a$a, $b$b, and $c$c,
| $a\left(b+c\right)$a(b+c) | $=$= | $ab+ac$ab+ac |
| and | ||
| $a\left(b-c\right)$a(b−c) | $=$= | $ab-ac$ab−ac |
This is known as the distributive property.
For example,
| $5\left(x+6\right)$5(x+6) | $=$= | $5x+5\times6$5x+5×6 |
| $=$= | $5x+30$5x+30 |
Since the distributive property is true for all numbers, we can also apply it to expressions with negative integers and rational numbers. Let's walk through a few examples.
Worked examples
Question 1
Distribute the expression $-6\left(x+4\right)$−6(x+4).
Think: We can apply the distributive property to multiply the expressions $-6$−6 and $x+4$x+4.
Do:
| $-6\left(x+4\right)$−6(x+4) | $=$= | $-6x+\left(-6\times4\right)$−6x+(−6×4) |
Apply the distributive property |
| $=$= | $-6x+\left(-24\right)$−6x+(−24) |
|
|
| $=$= | $-6x-24$−6x−24 |
Simplify |
Therefore the expression $-6\left(x+4\right)$−6(x+4) is equal to $-6x-24$−6x−24.
Reflect: Notice that the "$-$−" sign was also distributed. That is because we distributed the whole expression before the parenthesis.
Question 2
Distribute the expression $-2\left(3x-1\right)$−2(3x−1).
Think: We can apply the distributive property to multiply the expressions $-2$−2 and $3x-1$3x−1.
Do:
| $-2\left(3x-1\right)$−2(3x−1) | $=$= | $-2\times3x+\left(-2\times\left(-1\right)\right)$−2×3x+(−2×(−1)) |
Apply the distributive property |
| $=$= | $-6x+2$−6x+2 |
Remember the sign rules for multiplying integers: $-2\times3=-6$−2×3=−6 and
|
Therefore the expression $-2\left(3x-1\right)$−2(3x−1) is equal to $-6x+2$−6x+2.
Question 3
Distribute the expression $-2\left(6+x\right)$−2(6+x).
Think: We can apply the distributive property to multiply the expressions $-2$−2 and $6+x$6+x.
Do:
| $-2\left(6+x\right)$−2(6+x) | $=$= | $-2\times6+\left(-2\times x\right)$−2×6+(−2×x) |
Apply the distributive property |
| $=$= | $-12+\left(-2x\right)$−12+(−2x) |
Remember the sign rules for multiplying integers: $-2\times6=-12$−2×6=−12 and |
|
| $=$= | $-12-2x$−12−2x |
Simplify |
Therefore the expression $-2\left(6+x\right)$−2(6+x) is equal to $-12-2x$−12−2x.
Practice questions
Question 4
Distribute the expression $4\left(t+6\right)$4(t+6).
Question 5
Distribute the expression $-\left(5-a\right)$−(5−a).
Question 6
A student incorrectly used the distributive property and wrote $7\left(4x+3\right)=28x+3$7(4x+3)=28x+3.
Which of the following is the best explanation to help the student correct their error?
They have multiplied $4x$4x and $7$7 rather than adding them.
AThey have forgotten to multiply the second part of the sum, $3$3, by the number outside the parentheses, $7$7.
BThey have added $4x$4x and $7$7 rather than multiplying them.
CThey have have multiplied the wrong term in the sum by $7$7. They should multiply $3$3, instead of $4x$4x, by $7$7.
D