Chapter 9: Adding and Subtracting with Variables 115How about this one? Simplify -4xy(6x + 7y) using the same rules. Parentheses first. But this time,
you can’t add what’s in the parentheses because they’re unlike terms. Don’t panic. You can only
do what you can do, so just move on. There are no exponents, so it’s time to multiply, and for
that, you’ll need the distributive property.
-4xy(6x + 7y) = -4xy(6x) + (-4xy)(7y)
Now, as you look at each multiplication, multiply coefficients and combine what you can. Don’t
be afraid to rearrange. Remember multiplication is commutative and associative.
-4xy(6x) = -4 × 6 × x × x × y = -24x^2 y
(- 4xy)(7y) = -4 × 7 × x × y × y = -28xy^2
Put the pieces back together to get
-4xy(6x + 7y) = -4xy(6x) + (-4xy)(7y) = -24x^2 y + -28xy^2
Because you have the plus from the addition problem followed immediately by the minus sign
from the -28, you can write your answer as -24x^2 y – 28xy^2.
Your basic rules are:
- Combine what’s in the parentheses if you can.
- Distribute multiplication over the addition or subtraction if you can’t combine the unlike
terms. - Simplify each multiplication.
- Check the signs.
Whenever you have negatives in problems like these, it’s important to be careful about the rules
for signs. If the multiplier that you’re distributing is positive, the signs aren’t usually a problem.
Terms in the parentheses that were positive will produce positive terms in the answer, and terms
that were negative in the parentheses will produce negative terms in the answer. Here are some
examples.
8 x(-3a + 2b) = 8 × (-3) × x × a + 8 × 2 × x × b = -24ax + 16bx
5 t(7t^2 – 3t) = 5 × 7 × t × t^2 – 5 × 3 × t × t = 35t^3 – 15t^2
When the multiplier is negative, however, you need to work carefully and remember that if you
distribute a negative term, the signs of each term in the parentheses will change. Let’s take those
two examples and make the multipliers negative to see what happens.
Take the first example and change 8x to -8x. Distribute the -8x.
-8x(-3a + 2b) = -8 × (-3) × x × a + (-8) × 2 × x × b