Multiplying Polynomials; Special Products5.55.5

1. Multiply a polynomial by a monomial.2. Multiply binomials.3. Multiply polynomials.4. Determine the product when given special polynomial

factors.

Objective 1

Multiply a polynomial by a monomial.

Multiply. 22 6 2 1 p p p

22 6 2 1 p p p

3 212 4 2 p p p

2p • 6p2

2p • 2p2p • –1

2p ∙ 6p2 + 2p ∙ 2p + 2p ∙ – 1

Multiplying a Polynomial by a MonomialUse the distributive property to multiply each term in the polynomial by the monomial.

22 6 2 1 p p p 3 212 4 2 p p p

When an equation in one variable is solved the answer is a point on a line.

Multiply:Multiply:

625 2 aaa

1

101

51

201

5 2342 rrrrr

2 3 3 42 3 5 4 a b a b ab a bc

aaa 30510 23

cbabababa 2264325 81026

23456 521

41

5 rrrrr

Objective 2

Multiply binomials.

Multiply. 7 3x x

7 3x x

7 • x

7 • 3

x • 3

x • x

2= 3 7 21 x x x 2 10 = 21x x

xx 3 x x 7 37

Multiplying Polynomials 1. Multiply every term in the second polynomial by every term in the first polynomial.

2. Combine like terms.

7 3x x 2 10 = 21x x

Multiply. 2 1 5 x x

FOIL: First Outside Inside LastFOIL: First Outside Inside Last

2 1 5 x x

First Outside Inside Last

1 • xInside

1 • (–5)Last

2x • (–5)Outside

2x • x

First

22 10 5x x x 22 9 5x x 22 9 5x x

xx 2 52 x x 1 51

When an equation in one variable is solved the answer is a point on a line.

Multiply:Multiply:

57 nn 94 aa

1372 tt 3 5 2 7 x x

36132 aa 3522 nn

35116 2 xx 7236 2 tt

The product of two binomials can be shown in terms of geometry.

355x

7xx2

Length • width = Sum of the areas of the four internal rectangles

7 5 x x 2 5 7 35 x x x2 12 35 x x Combine like terms.

5

x

x 7

Objective 3

Multiply polynomials.

Multiply. 23 2 3 3 x x x

23 2 3 3 x x x

3 2 2 2 3 3 6 9 9 x x x x x3 2 2 3 6 9x x x

(–3) • 2x2

(–3) • 3x(–3) • 3

x • 3xx • 3

Horizontal Multiplication

332 2 xxxxx 333323 2 xx

Multiply. 23 2 3 3 x x xVertical Multiplication

332 2 xx

3x26x

x323x32x23x

x9

x6

9

32x 9

When an equation in one variable is solved the answer is a point on a line.

Multiply:Multiply:

233 2 xxx

22 93232 gfgfgf

67

693

23

3

2

23

xx

xx

xxx

323 27274 gfgf

Objective 4

Determine the product when given special polynomial factors.

When an equation in one variable is solved the answer is a point on a line.

Multiply:Multiply:

55 nn 44 aa

7272 tt cbcb 5454

162 a 252 n

494 2 t22 2516 cb

Multiplying ConjugatesIf a and b are real numbers, variables, or expressions, then (a + b)(a – b) = a2 – b2.

Conjugates: Binomials that differ only in the sign separating the terms.

x + 9 and x – 9 2x + 3 and 2x – 3

–6x + 5 and –6x – 5

When an equation in one variable is solved the answer is a point on a line.

Multiply:Multiply:

55 nn 44 aa

7272 tt cbcb 5454

24a 25n

272 t 254 cb

1682 aa 25102 nn

49284 2 tt 22 254016 cbcb

2 ∙ 4a 2 ∙ – 5 n

Squaring a BinomialIf a and b are real numbers, variables, or expressions, then

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

(a – b)2 = a2 – 2ab + b2

or

rewrite and foil !!rewrite and foil !!

Remember the shortcut

When an equation in one variable is solved the answer is a point on a line.

Multiply:Multiply:

229 k 25y

214 t 276 c

25102 yy 43681 2 kk

1816 2 tt2498436 cc

Slide 5- 22Copyright © 2011 Pearson Education, Inc.

Multiply.

a)

b)

c)

d)

2 5 3 y y

22 15 y y

22 2 15 y y

22 15 y y

22 11 15 y y

5.5

Slide 5- 23Copyright © 2011 Pearson Education, Inc.

Multiply.

a)

b)

c)

d)

2 5 3 y y

22 15 y y

22 2 15 y y

22 15 y y

22 11 15 y y

5.5

Slide 5- 24Copyright © 2011 Pearson Education, Inc.

Multiply.

a)

b)

c)

d)

25 4x

225 40 8 x x

225 20 8 x x

225 20 16 x x

225 40 16 x x

5.5

Slide 5- 25Copyright © 2011 Pearson Education, Inc.

Multiply.

a)

b)

c)

d)

25 4x

225 40 8 x x

225 20 8 x x

225 20 16 x x

225 40 16 x x

5.5