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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