Download Warm-Up Exercises

Warm-Up Exercises
Simplify the expression.
1. (–3x3)(5x)
ANSWER
–15x4
2. 9x – 18x
ANSWER
–9x
3. 10y2 + 7y – 8y2 – 1
ANSWER
2y2 + 7y – 1
Warm-Up Exercises
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.
Warm-Up1Exercises
EXAMPLE
Add polynomials vertically and horizontally
a. Add 2x3 – 5x2 + 3x – 9 and x3 + 6x2 + 11 in a vertical
format.
SOLUTION
a.
2x3 – 5x2 + 3x – 9
+ x3 + 6x2
+ 11
3x3 + x2 + 3x + 2
Warm-Up1Exercises
EXAMPLE
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
Warm-Up2Exercises
EXAMPLE
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
8x3 – x2 – 5x + 1
3x3 + 2x2 – x + 7
+ – 3x3 – 2x2 + x – 7
5x3 – 3x2 – 4x – 6
Warm-Up2Exercises
EXAMPLE
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
Warm-Up
Exercises
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
Warm-Up
Exercises
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 3 + 13d 2 + 4
= 9d 3 –3 d 3 + 13d 2 + 8d – 3 + 4
= 8d 3 + 13d 2 + 8d + 1
Warm-Up3Exercises
EXAMPLE
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
– 2y2 + 3y – 6
a.
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.
Warm-Up3Exercises
EXAMPLE
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
Warm-Up4Exercises
EXAMPLE
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
Warm-Up5Exercises
EXAMPLE
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
= 64x2 – 48x + 9
Square of a binomial
Warm-Up Exercises
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
Warm-Up Exercises
Expanding a binomial using Pascal’s Triangle
Pascal’s Triangle
1
1 1
1 2 1
1
1
1
1
3
4
5
6
3
6
10
15
1
4
10
20
1
5
15
Write the
next row.
1
6
1
Expand (pq + 5)3 = (pq)3 + 3(pq)2(5) + 3(pq)(5)2 + 53 Cube of a
binomial
= p3q3 + 15p2q2 + 75pq + 125
Warm-Up Exercises
From Pascal’s triangle write down
the 4th row.
Expand (x + 3)4
1
4
6
4
1
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.
Warm-Up
Exercises
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
3x3 – x2 – 5x
3x3 + 5x2 – 7x – 10
Multiply 3x2 – x – 5 by 2 .
Multiply 3x2 – x – 5 by x .
Combine like terms.
Warm-Up
Exercises
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)
Warm-Up
Exercises
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
Warm-Up6Exercises
EXAMPLE
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 = – 0.575t2 + 10.9t + 548 and O = – 0.249t + 15.4
where t is the number of years since 1980. 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?
Warm-Up6Exercises
EXAMPLE
Use polynomial models
SOLUTION
To find a model for T, multiply the two given models.
– 0.575t2 +
10.9t +
– 0.249t +
548
15.4
– 8.855t2 + 167.86t + 8439.2
0.143175t3 – 2.7141t2 – 136.452t
0.143175t3 – 11.5691t2 + 31.408t + 8439.2
Warm-Up6Exercises
EXAMPLE
Use polynomial models
ANSWER
Total daily oil output can be modeled by
T = 0.143t3 – 11.6t2 + 31.4t + 8440 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.
Warm-Up
Exercises
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
1980. Write a model for the average total cost T of
drilling a new well.
D = 109t + 4010
C = 0.542t2 – 7.16t + 79.4
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 6
SOLUTION
To find a model for T, multiply the two given models.
0.542t2 – 7.16t + 79.4
109t + 4010
2173.68t2 + 28711.6t + 318394
59.078t3 – 780.44t2 – 8654.6t
59.078t3 + 1392.98t2 – 20057t + 318394
Warm-Up
Exercises
GUIDED
PRACTICE
ANSWER
for Example 6
Total daily oil output can be modeled by
T = 59.078t3 + 1392.98t2 – 20,057t + 318394.
Warm-Up
Exercises
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)
Warm-Up
Exercises
Daily
Homework
Quiz
3. (5a2 + 6a + 9)(2a – 3)
ANSWER
8a3 – 27
4. 6(x – 1)(x + 1)
ANSWER
6x2 – 6
Warm-Up
Exercises
Daily
Homework
Quiz
5. The floor space in square feet of retail stores
A, B and C can be modeled by
A = x2 + 4x – 7, B = 2x2 – 7x + 1 and C = – 4x2 + 250x – 1.
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?
ANSWER
– x2 + 247x – 7; 9843 ft2.