116 Part 2: Into the Unknown
Now that first term has a negative multiplied by a negative, so the result is going to be positive.
The second term has one negative, so that result will be negative.
-8x(-3a + 2b) = -8 × (-3) × x × a + (-8) × 2 × x × b = +24ax + -16bx
You can eliminate the plus signs, condensing to 24ax – 16x.
If you change 5t to -5t in the second example and distribute, you get -5t(7t^2 – 3t) =
-5 × 7 × t × t^2 – (-5) × 3 × t × t.
The first term will come out negative, but the second term is where you have to be careful. The
product (-5) × 3 × t × t will give you a negative result, -15t^2 , but you also have a minus in front
of that, the one that was connecting the original terms. You get a product of -35t^3 – -15t^2 , but
the double minus becomes a plus so you end up with -35t^3 + 15t^2. The minus on the multiplier
switches the signs.
Sometimes you’ll find an expression with more than one set of parentheses and lots of terms,
some like and some unlike, and some that start out unlike and then turn into like terms. You just
need to take things step by step and pay attention to what is happening. Here’s an example.
-3x(6x^2 – 5) + 8x^2 (4x – 9)
There are no like terms in the first parentheses, so you can’t do anything there, and no like terms
in the second set of parentheses, either. Move on to multiplying. Use the distributive property.
-3x(6x^2 – 5) + 8x^2 (4x – 9)
= -3 × 6 × x × x^2 – (-3) × 5 × x + 8x^2 (4x – 9)
= -3 × 6 × x × x^2 – (-3) × 5 × x + 8 × 4 × x^2 × x – 8 × 9 × x^2
= -18x^3 + 15x + 32x^3 – 72x^2
Now notice you have like terms: -18x^3 and 32x^3. You can combine those to get a final answer of
14 x^3 + 15x -72x^2. It’s traditional to put your terms in order from highest exponent to lowest, so
rewrite it as 14x^3 – 72x^2 + 15x.CHECK POINT
Simplify each expression.
16. 6x(2x + 9)- 12 + 5(x + 1)
- 6t^2 (t – 3) – 2t^2
19. 5y(6y + 2) + 7y^2 (4 – 12y)
20. 8a(2b – 5) – 2b(a – 2)