Post on 02-Jan-2016
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Warm-Up Exercises
3. The number of hardback h and paperback p books (in hundreds) sold from 1999–2005 can be modeled by h = 0.2t2 – 1.7t + 14 and p = 0.17t3 – 2.7t2 + 11.7t + 27 where t is the number of years since 1999. About how many books sold in 2003?
ANSWER 5200
Warm-Up ExercisesEXAMPLE 1 Multiply a monomial and a polynomial
Find the product 2x3(x3 + 3x2 – 2x + 5).
2x3(x3 + 3x2 – 2x + 5) Write product.
= 2x3(x3) + 2x3(3x2) – 2x3(2x) + 2x3(5) Distributive property
= 2x6 + 6x5 – 4x4 + 10x3 Product of powers property
Warm-Up ExercisesMultiply polynomials using a tableEXAMPLE 2
Find the product (x – 4)(3x + 2).
STEP 1 Write subtraction as addition in each polynomial.
(x – 4)(3x + 2) = [x + (– 4)](3x + 2)
SOLUTION
Warm-Up ExercisesMultiply polynomials using a tableEXAMPLE 2
ANSWER
The product is 3x2 + 2x – 12x – 8, or 3x2 – 10x – 8.
3x2x
– 4
3x 2
– 8– 12x
2x3x2x
– 4
3x 2
STEP 2
Make a table of products.
Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2
Find the product.
1. x(7x2 +4) ANSWER 7x3 + 4x
2. (a +3)(2a +1) ANSWER 2a2 + 7a + 3
3. (4n – 1)(n + 5) ANSWER 4n2 + 19n – 5
Warm-Up ExercisesEXAMPLE 3 Multiply polynomials vertically
Find the product (b2 + 6b – 7)(3b – 4).
SOLUTION
STEP 1
Multiply by – 4.
b2 + 6b – 7
– 4b2 – 24b + 28
3b – 4
STEP 2Multiply by 3b.
b2 + 6b – 7
3b – 4
– 4b2 – 24b + 28
3b3 + 18b2 – 21b
Warm-Up ExercisesMultiply polynomials vertically
EXAMPLE 3
STEP 3
Add products.
b2 + 6b – 7
3b – 4
– 4b2 – 24b + 28
3b3 + 18b2 – 21b
3b3 + 14b2 – 45b + 28
Warm-Up ExercisesMultiply polynomials horizontallyEXAMPLE 4
Find the product (2x2 + 5x – 1)(4x – 3).
(2x2 + 5x – 1)(4x – 3) Write product.
= 2x2(4x – 3) + 5x(4x – 3) – 1(4x – 3)
= 8x3 – 6x2 + 20x2 – 15x – 4x + 3
Distributive property
Distributive property
= 8x3 + 14x2 – 19x + 3 Combine like terms.
FOIL PATTERN The letters of the word FOIL can helpyou to remember how to use the distributive property tomultiply binomials. The letters should remind you of thewords First, Outer, Inner, and Last.
Warm-Up ExercisesMultiply polynomials horizontally
EXAMPLE 4
(2x + 3)(4x + 1)
First Outer Inner Last
= 8x2 + 2x + 12x + 3
Warm-Up ExercisesMultiply binomials using the FOIL pattern
EXAMPLE 5
Find the product (3a + 4)(a – 2).
(3a + 4)(a – 2)
= (3a)(a) + (3a)(– 2) + (4)(a) + (4)(– 2) Write products of terms.
= 3a2 + (– 6a) + 4a + (– 8) Multiply.
= 3a2 – 2a – 8 Combine like terms.
Warm-Up ExercisesGUIDED PRACTICE for Examples 3, 4, and 5
Find the product.
(x2 + 2x +1)(x + 2)4. x3 + 4x2 + 5x + 2ANSWER
5. (3y2 –y + 5)(2y – 3) ANSWER 6y3 – 11y2 + 13y – 15
6. (4b –5)(b – 2) ANSWER 4b2 – 13b + 10
Warm-Up Exercises
SOLUTION
EXAMPLE 6 Standardized Test Practice
The dimensions of a rectangle are x + 3 and x + 2. Which expression represents the area of the rectangle?
Area = length width Formula for area of a rectangle
= (x + 3)(x + 2) Substitute for length and width.
= x2 + 2x + 3x + 6 Multiply binomials.
x2 + 6 A x2 + 6xDx2 + 5x + 6B x2 + 6x + 6C
Warm-Up ExercisesEXAMPLE 6
= x2 + 5x + 6 Combine like terms.
ANSWER
The correct answer is B. A DCB
Standardized Test Practice
CHECKYou can use a graph to check your answer. Use a graphing calculator to display the graphs of y1 = (x + 3)(x + 2) and y2 = x2 + 5x + 6 in the same viewing window. Because the graphs coincide, you know that the product of x + 3 and x + 2 is x2 + 5x + 6.
Warm-Up Exercises
SKATEBOARDING
EXAMPLE 7 Solve a multi-step problem
You are designing a rectangular skateboard park on a lot that is on the corner of a city block. The park will have a walkway along two sides. The dimensions of the lot and the walkway are shown in the diagram.
Write a polynomial that represents the area of the skateboard park.
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What is the area of the park if the walkway is 3 feet wide?
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Warm-Up Exercises
SOLUTION
EXAMPLE 7 Solve a multi-step problem
STEP 1Write a polynomial using the formula for the area of a rectangle. The length is 45 – x. The width is 33 – x.
Area = length width Formula for area of a rectangle
= (45 – x)(33 – x) Substitute for length and width.
= 1485 – 45x – 33x + x2 Multiply binomials.
= 1485 – 78x + x2 Combine like terms.
Warm-Up ExercisesEXAMPLE 7 Solve a multi-step problem
Substitute 3 for x and evaluate.
STEP 2
Area = 1485 – 78(3) + (3)2 = 1260
ANSWER
The area of the park is 1260 square feet.
Warm-Up ExercisesGUIDED PRACTICE for Examples 6 and 7
x2 + 45x A
x2 + 45x + 45D
x2 + 45B
x2 + 14x + 45C
The dimensions of a rectangle are x + 5 and x + 9. Which expression represents the area of the rectangle?
7
ANSWER C
C
Warm-Up ExercisesGUIDED PRACTICE for Examples 6 and 7
You are planning to build a walkway that surrounds a rectangular garden, as shown. The width of the walkway around the garden is the same on every side.
8. GARDEN DESIGN
Warm-Up ExercisesGUIDED PRACTICE for Examples 6 and 7
a. Write a polynomial that represents the combined area of the garden and the walkway.
b. Find the combined area when the width of the walkway is 4 feet.
4x2 + 38x + 90ANSWER
306 ft2ANSWER
Warm-Up ExercisesDaily Homework Quiz
Find the product.
1. 3x(x3 – 3x2 + 2x – 4)
3x4 – 9x3 + 6x2 – 12x ANSWER
2. (y – 4)(2y + 5)
ANSWER 2y2 – 3y – 20
Warm-Up ExercisesDaily Homework Quiz
3. (4x + 3)(3x – 2)
ANSWER 12x2 + x – 6
4. (b2 – 2b – 1)(3b – 5)
ANSWER 3b3 – 11b2 + 7b + 5