Section 12.6 Special Products

1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Exponents and
Polynomials
12

2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
1. Square binomials.
2. Find the product of the sum and difference
of two terms.
3. Find greater powers of binomials.
Objectives
12.6 Special Products

3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Square of a Binomial
The square of a binomial is a trinomial consisting of the
square of the first term, plus twice the product of the two
terms, plus the square of the last term of the binomial.
For a and b,
(x + y)2 = x2 + 2xy + y2.
Also, (x – y)2 = x2 – 2xy + y2.
Square Binomials

4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
(a) (4x + 5)2
Square Binomials
Example
Square each binomial.
x y
= (4x)2 + 2(4x)(5) + 52
x2 x y y2
= 16x2 + 40x + 25
(b) (6x – 2y)2 = (6x)2 – 2(6x)(2y) + (2y)2 = 36x2 – 24xy + 4y2

5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Find the Product of the Sum and Difference of Two Terms
Product of the Sum and Difference of Two Terms
(x + y)(x – y) = x2 – y2

6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
(a) (3m + 8n)(3m – 8n)
Find the Product of the Sum and Difference of Two Terms
= (3m)2 – (8n)2
= 9m2 – 64n2
(b) 2p(5p + 1)(5p – 1) = 2p[(5p)2 – (1)2]
= 2p(25p2 – 1)
= 50p3 – 2p
Example
Find each product.

7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Find Greater Powers of Binomials
(a) (7 – 4s)3 = (7 – 4s)(7 – 4s)2
= (7 – 4s)[72 – 2(7)(4s) + (4s)2]
= 343 – 588s + 336s2 – 64s3
Example
Find each product.
= (7 – 4s)(49 – 56s + 16s2)
= 343 – 392s + 112s2 – 196s + 224s2 – 64s3

8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Find Greater Powers of Binomials
(b) (2t + 3v)4 = (2t + 3v)2(2t + 3v)2
= [(2t)2 + 2(2t)(3v) + (3v)2][(2t)2 + 2(2t)(3v) + (3v)2]
Example (cont)
Find each product.
= (4t2 + 12tv + 9v2)(4t2 + 12tv + 9v2)
= 16t4 + 48t3v + 36t2v2 + 48t3v + 144t2v2 + 108tv3
+ 36t2v2 + 108tv3 + 81v4
= 16t4 + 96t3v + 216t2v2 + 216tv3 + 81v4