1. Simplify the expression.
1. (–3x3)(5x)
ANSWER
–15x4
2. 9x – 18x
ANSWER
–9x
y2 + 7y – 8y2 – 1
ANSWER
2y2 + 7y – 1

2. Simplify the expression.
4. 4(– 5a + 6) –2(a – 8)
ANSWER –4
5. Each side of a square is (2x + 5) inches long Write an expression for the perimeter of the square.
ANSWER
(8x + 20) in.

3. EXAMPLE 1
Add polynomials vertically and horizontally
a. Add 2x3 – 5x2 + 3x – 9 and x3 + 6x in a vertical format.
SOLUTION
a x3 – 5x2 + 3x – 9
+ x3 + 6x
3x3 + x2 + 3x + 2

4. EXAMPLE 1
Add polynomials vertically and horizontally
b. Add 3y3 – 2y2 – 7y and – 4y2 + 2y – 5 in a horizontal
format.
(3y3 – 2y2 – 7y) + (– 4y2 + 2y – 5)
= 3y3 – 2y2 – 4y2 – 7y + 2y – 5
= 3y3 – 6y2 – 5y – 5

5. EXAMPLE 2
Subtract polynomials vertically and horizontally
a. Subtract 3x3 + 2x2 – x + 7 from 8x3 – x2 – 5x + 1 in a vertical format.
SOLUTION
a. Align like terms, then draw your line and change your signs (of the bottom row and add straight down).
8x3 – x2 – 5x + 1
3x3 + 2x2 – x + 7
8x3 – x2 – 5x + 1
+ – 3x3 – 2x2 + x – 7
5x3 – 3x2 – 4x – 6

6. EXAMPLE 2
Subtract polynomials vertically and horizontally
b. Subtract 5z2 – z + 3 from 4z2 + 9z – 12 in a horizontal
format.
Distribute the negative sign to ALL of the
subtracted polynomial, then add like terms.
(4z2 + 9z – 12) – (5z2 – z + 3) = 4z2 + 9z – 12 – 5z2 + z – 3
= 4z2 – 5z2 + 9z + z – 12 – 3
= – z2 + 10z – 15

7. GUIDED PRACTICE
for Examples 1 and 2
Find the sum or difference.
1. (t2 – 6t + 2) + (5t2 – t – 8)
SOLUTION
t2 – 6t + 2
+ 5t2 – t – 8
6t2 – 7t – 6

8. GUIDED PRACTICE
for Examples 1 and 2
2. (8d – 3 + 9d 3) – (d 3 – 13d 2 – 4)
SOLUTION
= (8d – 3 + 9d 3) – (d 3 – 13d 2 – 4)
= (8d – 3 + 9d 3) – d d 2 + 4
= 9d 3 –3 d d 2 + 8d – 3 + 4
= 8d d 2 + 8d + 1

9. Multiply polynomials vertically and horizontally
EXAMPLE 3
Multiply polynomials vertically and horizontally
a. Multiply – 2y2 + 3y – 6 and y – 2 in a vertical format.
b. Multiply x + 3 and 3x2 – 2x + 4 in a horizontal format.
SOLUTION
a – 2y2 + 3y – 6
y – 2
4y2 – 6y + 12
Multiply – 2y2 + 3y – 6 by – 2 .
– 2y3 + 3y2 – 6y
Multiply – 2y2 + 3y – 6 by y
– 2y3 +7y2 –12y + 12
Combine like terms.

10. EXAMPLE 3
Multiply polynomials vertically and horizontally
b. (x + 3)(3x2 – 2x + 4) = (x + 3)3x2 – (x + 3)2x + (x + 3)4
= 3x3 + 9x2 – 2x2 – 6x + 4x + 12
= 3x3 + 7x2 – 2x + 12

11. EXAMPLE 4
Multiply three binomials
Multiply x – 5, x + 1, and x + 3 in a horizontal format.
(x – 5)(x + 1)(x + 3) = (x2 – 4x – 5)(x + 3)
= (x2 – 4x – 5)x + (x2 – 4x – 5)3
= x3 – 4x2 – 5x + 3x2 – 12x – 15
= x3 – x2 – 17x – 15

12. Use special product patterns
EXAMPLE 5
Use special product patterns
a. (3t + 4)(3t – 4)
= (3t)2 – 42
Sum and difference
= 9t2 – 16
b. (8x – 3)2
= (8x)2 – 2(8x)(3) + 32
Square of a binomial
= 64x2 – 48x + 9

13. The Binomial Theorem
Let’s look at the expansion of (x + y)n
(x + y)0 = 1
(x + y)1 = x + y
(x + y)2 = x2 +2xy + y2
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

14. Expanding a binomial using Pascal’s Triangle Pascal’s Triangle 1 1 2 1
Write the
next row.
Cube of a binomial
Expand (pq + 5)3
= (pq)3 + 3(pq)2(5) + 3(pq)(5)2 + 53
= p3q3 + 15p2q2 + 75pq + 125

15. From Pascal’s triangle write down
the 4th row.
Expand (x + 3)4
These numbers are the same numbers that are the
coefficients of the binomial expansion.
The expansion of (a + b)4 is:
1a4b0 + 4a3b1 + 6a2b2 + 4a1b3 + 1a0b4
Notice that the exponents always add up to 4 with
the a’s going in descending order and the b’s in
ascending order.
Now substitute x in for a and 3 in for b.

16. GUIDED PRACTICE for Examples 3, 4 and 5 Find the product.
3. (x + 2)(3x2 – x – 5)
SOLUTION
3x2 – x – 5
x + 2
6x2 – 2x – 10
Multiply 3x2 – x – 5 by 2 .
3x3 – x2 – 5x
Multiply 3x2 – x – 5 by x .
3x3 + 5x2 – 7x – 10
Combine like terms.

17. GUIDED PRACTICE
for Examples 3, 4 and 5
4. (a – 5)(a + 2)(a + 6)
SOLUTION
(a – 5)(a + 2)(a + 6)
= (a2 – 3a – 10)(a + 6)
= (a2 – 3a – 10)a + (a2 – 3a – 10)6
= (a3 – 3a2 – 10a + 6a2 – 18a – 60)
= (a3 + 3a2 – 28a – 60)

18. GUIDED PRACTICE
for Examples 3, 4 and 5
5. (xy – 4)3
SOLUTION
(xy – 4)3
= (xy)3 + 3(xy)2(– 4) + 3(xy)(– 4)2 + (– 4)3
= x3y3 – 12x2y2 + 48xy – 64

19. EXAMPLE 6
Use polynomial models
Petroleum
Since 1980, the number W (in thousands) of United States wells producing crude oil and the average daily oil output per well O (in barrels) can be modeled by
W = – 0.575t t and O = – 0.249t
where t is the number of years since Write a model for the average total amount T of crude oil produced per day (product of W and T). What was the average total amount of crude oil produced per day in 2000?

20. EXAMPLE 6
Use polynomial models
SOLUTION
To find a model for T, multiply the two given models.
– 0.575t t
– t
– t t
t3 – t2 – t
t3 – t t

21. EXAMPLE 6
Use polynomial models
Total daily oil output can be modeled by T = 0.143t3 – 11.6t t where T is measured in thousands of barrels. By substituting t = 20 into the model, you can estimate that the average total amount of crude oil produced per day in 2000 was about 5570 thousand barrels, or 5,570,000 barrels.
ANSWER

22. GUIDED PRACTICE
for Example 6
Industry 6.
The models below give the average depth D (in feet) of new wells drilled and the average cost per foot C (in dollars) of drilling a new well. In both models, t represents the number of years since Write a model for the average total cost T of drilling a new well.
D = 109t
C = 0.542t2 – 7.16t

23. GUIDED PRACTICE
for Example 6
SOLUTION
To find a model for T, multiply the two given models.
t t
0.542t2 – 7.16t
109t
59.078t3 – t2 – t
59.078t t2 – 20057t

24. GUIDED PRACTICE
for Example 6
Total daily oil output can be modeled by T = t t2 – 20,057t
ANSWER

25. Daily Homework Quiz
Find the sum, difference, or product. 1.
(3x2 + 5x + 2) + (x2 – 3x + 6)
ANSWER
4x2 + 2x + 8 2.
(5p3 + 2p2 – 3p – 7) - (2p3 – 4p2 – 5p + 6)
ANSWER
(3p3 + 6p2 + 2p – 13)

26. Daily Homework Quiz 3.
(5a2 + 6a + 9)(2a – 3)
ANSWER
8a3 – 27 4.
6(x – 1)(x + 1)
ANSWER
6x2 – 6

27. Daily Homework Quiz
5. The floor space in square feet of retail stores
A, B and C can be modeled by
and
Write a model for the total Amount of floor space
T for all three stores. What Is the total floor space
of the three stores if x = 50?
A = x2 + 4x – 7, B = 2x2 – 7x + 1
C = – 4x x – 1.
ANSWER
– x x – 7; 9843 ft2.