Holt Algebra 2 6-2 Multiplying Polynomials 6-2 Multiplying Polynomials Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz

Holt Algebra 2

6-2 Multiplying Polynomials6-2 Multiplying Polynomials

Holt Algebra 2

Warm Up

Lesson Presentation

Lesson Quiz

Holt Algebra 2

6-2 Multiplying Polynomials

Warm UpMultiply.

1. x(x3)

3. 2(5x3)

5. xy(7x2)

6. 3y2(–3y)

7x3y

x4

10x3

–9y3

2. 3x2(x5) 3x7

4. x(6x2) 6x3

Holt Algebra 2


Multiply polynomials.

Use binomial expansion to expand binomial expressions that are raised to positive integer powers.

Objectives

Holt Algebra 2


To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

Holt Algebra 2


Find each product.

Example 1: Multiplying a Monomial and a Polynomial

A. 4y2(y2 + 3)

Distribute.

B. fg(f4 + 2f3g – 3f2g2 + fg3)

4y2 y2 + 4y2 3

4y2(y2 + 3)

Multiply.4y4 + 12y2

fg(f4 + 2f3g – 3f2g2 + fg3)

Distribute.

Multiply.

fg f4 + fg 2f3g – fg 3f2g2 + fg fg3

f5g + 2f4g2 – 3f3g3 + f2g4

Holt Algebra 2


Check It Out! Example 1

Find each product.

a. 3cd2(4c2d – 6cd + 14cd2)

Distribute.

b. x2y(6y3 + y2 – 28y + 30)

3cd2 4c2d – 3cd2 6cd + 3cd2 14cd2

3cd2(4c2d – 6cd + 14cd2)

Multiply.12c3d3 – 18c2d3 + 42c2d4

x2y(6y3 + y2 – 28y + 30)

Distribute.

Multiply.

x2y 6y3 + x2y y2 – x2y 28y + x2y 30

6x2y4 + x2y3 – 28x2y2 + 30x2y

Holt Algebra 2


To multiply any two polynomials, use the Distributive Property and multiply each term in the second polynomial by each term in the first.

Keep in mind that if one polynomial has m terms and the other has n terms, then the product has mn terms before it is simplified.

Holt Algebra 2


Find the product.

Example 2A: Multiplying Polynomials

(a – 3)(2 – 5a + a2)

a(a2) + a(–5a) + a(2) – 3(a2) – 3(–5a) –3(2)

Method 1 Multiply horizontally.

a3 – 5a2 + 2a – 3a2 + 15a – 6

a3 – 8a2 + 17a – 6

Write polynomials in standard form.

Distribute a and then –3.

Multiply. Add exponents.

Combine like terms.

(a – 3)(a2 – 5a + 2)

Holt Algebra 2


(a – 3)(2 – 5a + a2)

Find the product.

Example 2A: Multiplying Polynomials

Method 2 Multiply vertically.

Write each polynomial in standard form.Multiply (a2 – 5a + 2) by –3.

a2 – 5a + 2 a – 3

– 3a2 + 15a – 6

Multiply (a2 – 5a + 2) by a, and align like terms.

a3 – 5a2 + 2a

a3 – 8a2 + 17a – 6 Combine like terms.

Holt Algebra 2


(y2 – 7y + 5)(y2 – y – 3) Find the product.

Example 2B: Multiplying Polynomials

Multiply each term of one polynomial by each term of the other. Use a table to organize the products.

y4 –y3 –3y2

–7y3 7y2 21y

5y2 –5y –15

y2 –y –3

y2

–7y

5

The top left corner is the first term in the product. Combine terms along diagonals to get the middle terms. The bottom right corner is the last term in the product.

y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15

y4 – 8y3 + 9y2 + 16y – 15

Holt Algebra 2


Check It Out! Example 2a

Find the product.

(3b – 2c)(3b2 – bc – 2c2)

3b(3b2) + 3b(–2c2) + 3b(–bc) – 2c(3b2) – 2c(–2c2) – 2c(–bc)

Multiply horizontally.

9b3 – 6bc2 – 3b2c – 6b2c + 4c3 + 2bc2

9b3 – 9b2c – 4bc2 + 4c3

Write polynomials in standard form.

Distribute 3b and then –2c.

Multiply. Add exponents.

Combine like terms.

(3b – 2c)(3b2 – 2c2 – bc)

Holt Algebra 2


(x2 – 4x + 1)(x2 + 5x – 2)

Find the product.Check It Out! Example 2b

Multiply each term of one polynomial by each term of the other. Use a table to organize the products.

x4 –4x3 x2

5x3 –20x2 5x

–2x2 8x –2

x2 –4x 1

x2

5x

–2

The top left corner is the first term in the product. Combine terms along diagonals to get the middle terms. The bottom right corner is the last term in the product.

x4 + (–4x3 + 5x3) + (–2x2 – 20x2 + x2) + (8x + 5x) – 2

x4 + x3 – 21x2 + 13x – 2

Holt Algebra 2


Example 3: Business Application

A standard Burly Box is p ft by 3p ft by 4p ft. A large Burly Box has 1.5 ft added to each dimension. Write a polynomial V(p) in standard form that can be used to find the volume of a large Burly Box.The volume of a large Burly Box is the product of the area of the base and height. V(p) = A(p) h(p)The area of the base of the large Burly Box is the product of the length and width of the box. A(p) = l(p) w(p)

The length, width, and height of the large Burly Box are greater than that of the standard Burly Box. l(p) = p + 1.5, w(p) = 3p + 1.5, h(p) = 4p + 1.5

Holt Algebra 2


p + 1.5

3p + 1.5

1.5p + 2.25

Solve A(p) = l(p) w(p).

3p2 + 4.5p

3p2 + 6p + 2.25

3p2 + 6p + 2.25

4p + 1.5

4.5p2 + 9p + 3.375

Solve V(p) = A(p) h(p).

12p3 + 24p2 + 9p

12p3 + 28.5p2 + 18p + 3.375

The volume of a large Burly Box can be modeled byV(p) = 12p3 + 28.5p2 + 18p + 3.375

Example 3: Business Application

Holt Algebra 2



Mr. Silva manages a manufacturing plant. From 1990 through 2005 the number of units produced (in thousands) can be modeled by N(x) = 0.02x2 + 0.2x + 3. The average cost per unit (in dollars) can be modeled by C(x) = –0.004x2 – 0.1x + 3. Write a polynomial T(x) that can be used to model the total costs.

Total cost is the product of the number of units and the cost per unit.

T(x) = N(x) C(x)

Holt Algebra 2


0.02x2 + 0.2x + 3

–0.004x2 – 0.1x + 3

0.06x2 + 0.6x + 9

Multiply the two polynomials.

–0.002x3 – 0.02x2 – 0.3x

–0.00008x4 – 0.0028x3 + 0.028x2 + 0.3x + 9

Mr. Silva’s total manufacturing costs, in thousands of dollars, can be modeled by T(x) = –0.00008x4 – 0.0028x3 + 0.028x2 + 0.3x + 9


–0.00008x4 – 0.0008x3 – 0.012x2

Holt Algebra 2


Example 4: Expanding a Power of a Binomial

Find the product.(a + 2b)3

(a + 2b)(a + 2b)(a + 2b) Write in expanded form.

(a + 2b)(a2 + 4ab + 4b2) Multiply the last two binomial factors.

a(a2) + a(4ab) + a(4b2) + 2b(a2) + 2b(4ab) + 2b(4b2)

Distribute a and then 2b.

a3 + 4a2b + 4ab2 + 2a2b + 8ab2 + 8b3 Multiply.

a3 + 6a2b + 12ab2 + 8b3 Combine like terms.

Holt Algebra 2

6-2 Multiplying PolynomialsCheck It Out! Example 4a

Find the product.(x + 4)4

(x + 4)(x + 4)(x + 4)(x + 4) Write in expanded form.

Multiply the last two binomial factors.

(x + 4)(x + 4)(x2 + 8x + 16)

Multiply the first two binomial factors.

(x2 + 8x + 16)(x2 + 8x + 16)

x2(x2) + x2(8x) + x2(16) + 8x(x2) + 8x(8x) + 8x(16) + 16(x2) + 16(8x) + 16(16)

Distribute x2 and then 8x and then 16.

Multiply.

x4 + 8x3 + 16x2 + 8x3 + 64x2 + 128x + 16x2 + 128x + 256

Combine like terms.x4 + 16x3 + 96x2 + 256x + 256

Holt Algebra 2

6-2 Multiplying PolynomialsCheck It Out! Example 4b

Find the product.(2x – 1)3

(2x – 1)(2x – 1)(2x – 1) Write in expanded form.

(2x – 1)(4x2 – 4x + 1) Multiply the last two binomial factors.

2x(4x2) + 2x(–4x) + 2x(1) – 1(4x2) – 1(–4x) – 1(1)

Distribute 2x and then –1.

8x3 – 8x2 + 2x – 4x2 + 4x – 1 Multiply.

8x3 – 12x2 + 6x – 1 Combine like terms.

Holt Algebra 2


Notice the coefficients of the variables in the final product of (a + b)3. these coefficients are the numbers from the third row of Pascal's triangle.

Each row of Pascal’s triangle gives the coefficients of the corresponding binomial expansion. The pattern in the table can be extended to apply to the expansion of any binomial of the form (a + b)n, where n is a whole number.

Holt Algebra 2


This information is formalized by the Binomial Theorem, which you will study further in Chapter 11.

Holt Algebra 2

6-2 Multiplying PolynomialsExample 5: Using Pascal’s Triangle to Expand Binomial

ExpressionsExpand each expression.

A. (k – 5)3

B. (6m – 8)3

1 3 3 1 Identify the coefficients for n = 3, or row 3.

[1(k)3(–5)0] + [3(k)2(–5)1] + [3(k)1(–5)2] + [1(k)0(–5)3]

k3 – 15k2 + 75k – 125


[1(6m)3(–8)0] + [3(6m)2(–8)1] + [3(6m)1(–8)2] + [1(6m)0(–8)3]

216m3 – 864m2 + 1152m – 512

Holt Algebra 2



Expand each expression.

a. (x + 2)3


[1(x)3(2)0] + [3(x)2(2)1] + [3(x)1(2)2] + [1(x)0(2)3]

x3 + 6x2 + 12x + 8

b. (x – 4)5

1 5 10 10 5 1 Identify the coefficients for n = 5, or row 5.

[1(x)5(–4)0] + [5(x)4(–4)1] + [10(x)3(–4)2] + [10(x)2(–4)3] + [5(x)1(–4)4] + [1(x)0(–4)5]

x5 – 20x4 + 160x3 – 640x2 + 1280x – 1024

Holt Algebra 2



c. (3x + 1)4

1 4 6 4 1 Identify the coefficients for n = 4, or row 4.

[1(3x)4(1)0] + [4(3x)3(1)1] + [6(3x)2(1)2] + [4(3x)1(1)3] + [1(3x)0(1)4]

81x4 + 108x3 + 54x2 + 12x + 1

Expand the expression.

Holt Algebra 2


4. Find the product. (y – 5)4

Lesson Quiz

2. (2a3 – a + 3)(a2 + 3a – 5) 5jk2 – 10j2k1. 5jk(k – 2j)

2a5 + 6a4 – 11a3 + 14a – 15

y4 – 20y3 + 150y2 – 500y + 625

–0.03x4 – 0.1x3 + 1.27x2 – 0.1x + 10

3. The number of items is modeled by 0.3x2 + 0.1x + 2, and the cost per item is modeled by g(x) = –0.1x2 – 0.3x + 5. Write a polynomial c(x) that can be used to model the total cost.

Find each product.

5. Expand the expression. (3a – b)3 27a3 – 27a2b + 9ab2 – b3

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Holt Algebra 2 6-2 Multiplying Polynomials 6-2 Multiplying Polynomials Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz