HOMEWORK 15
RICKY NG
Question 15.1. Multiply (x + 2)(x + 10), from highest to lowest degree.
Solution. Let’s FOIL it out:
· 10} + |{z}
2 · x + |2 {z
· 10}
(x + 2)(x + 10) = x
· x+x
| {z
|{z}
F
2
O
I
L
= x + 10x + 2x + 20
= x2 + 12x + 20.
Question 15.2. Multiply (x + 9)(x + −9), from highest to lowest degree.
Solution. Again we can FOIL it:
(x + 9)(x − 9) = x2 − 9x + 9x − 81 = x2 − 81.
Question 15.3. Multiply (x − 4)(3x − 2), from highest to lowest degree.
Solution. FOIL again:
· −2}
(x − 4)(3x − 2) = x
· 3x} + x
| {z
| {z
| ·{z−2} + |−4{z· 3x} + −4
F
2
O
I
L
= 3x − 2x − 12x + 8
= 3x2 − 14x + 8.
Question 15.4. Multiply (2x5 + x2 )(5 − 3x4 ), from highest to lowest degree.
Solution. FOIL again:
5
2
2
(2x5 + x2 )(5 − 3x4 ) = |2x{z
· 5} + |2x5 ·{z−3x}4 + x
−3x}4
| {z· 5} + x
| · {z
F
5
I
O
9
2
= 10x − 6x + 5x − 3x
L
6
= −6x9 − 3x6 + 10x5 + 5x2 .
1
Question 15.5. Simplify −7x2 − (2x2 + 4x − 9), from highest to lowest degree.
Solution. Note that the minus sign − should be distributed into each of the terms inside
the parentheses!
−7x2 −(2x2 + 4x − 9) = −7x2 −2x2 −4x+9
= −9x2 − 4x + 9.
Question 15.6. Simplify 7x + 5(x − x3 ), from highest to lowest degree.
Solution. We should follow the order of operations; that is, in this case multiplication before addition. First we need to distribute the 5 into the parentheses.
7x + 5(x − x3 ) = 7x + 5cdotx − 5x3
= 7x+5x − 5x3
= 12x − 5x3
= −5x3 + 12x.
Question 15.7. Simplify (x − 2) − (x + 8), from highest to lowest degree.
Solution. Again, we need to distribute the − first.
(x − 2) − (x + 8) = x − 2−x−8
= x − x − 2 − 8 = −10.
Question 15.8. Simplify (5x2 − 7x + 2) + (x2 + 4x − 3), from highest to lowest degree.
Solution. Since we are just adding, we can drop the parentheses without any sign
changing.
(5x2 − 7x + 2) + (x2 + 4x − 3) = 5x2 − 7x + 2 + x2 + 4x − 3
= |5x2{z
+ x}2 −7x
| {z+ 4x} +2
| {z− 3}
x2 -term
2
x-term
constant-term
= 6x − 3x − 1.
2
Question 15.9. Simplify (x7 − 4x4 + 2x) − (6x5 − 3x4 − 5x), from highest to lowest
degree.
Solution. Distribute the − before dropping the parentheses:
(x7 − 4x4 + 2x)−(6x5 − 3x4 − 5x) = x7 − 4x4 + 2x−6x5 +3x4 +5x
4
4
= x7 − 6x5 −4x
| {z+ 5x}
{z+ 3x} +2x
|
7
5
x4 -term
4
x-term
= x − 6x − x + 7x.
Question 15.10. Simplify (x + 2)(x − 3)2 , from highest to lowest degree.
Solution. Follow the order of operation; in this case, first evaluate the exponent, then
multiply. Let’s first consider (x − 3)2 using FOIL:
(x − 3)2 = (x − 3)(x − 3)
= x2 − 3x − 3x + 9
= x2 − 6x + 9.
Hence,
(x + 2)(x − 3)2 = (x + 2)(x2 − 6x + 9).
By distributing the (x + 2) over the second polynomial, we get
(x + 2)(x2 − 6x + 9) = x(x2 − 6x + 9) + 2(x2 − 6x + 9)
= (x3 − 6x2 + 9x) + (2x2 − 12x + 18)
2
2
= x3 −6x
− 12x} +18
|
{z+ 2x} +9x
| {z
3
x2 -term
2
x-term
= x − 4x − 3x + 18.
3