1. 13
Exponents and Polynomials

2. 13.4 Special Products
Objectives
1. Square binomials. 2. Find the product of the sum and difference of two terms. 3. Find greater powers of binomials.

3. Square Binomials
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,
(a + b)2 = a2 + 2ab + b2.
Also, (a – b)2 = a2 – 2ab + b2.

4. Square Binomials Example 2 Square each binomial. (a) (4x + 5)2
= (4x)2 + 2(4x)(5) + 52
= 16x2 + 40x + 25 a b a2 a b b2
(b) (6x – 2y)2
= (6x)2 – 2(6x)(2y) + (2y)2
= 36x2 – 24xy + 4y2 4

5. Squaring Binomials CAUTION
A common error when squaring a binomial is to forget the middle term of the product. In general, 5

6. Find the Product of the Sum and Difference of Two Terms
(a + b)(a – b) = a2 – b2
Note
The expressions a + b and a – b, the sum and difference
of the same two terms, are called conjugates.
In the example above, x + 2 and x – 2 are conjugates.

7. Find the Product of the Sum and Difference of Two Terms
Example 4
Find each product.
(a) (3m + 8n)(3m – 8n)
= (3m)2 – (8n)2
= 9m2 – 64n2
(b) 2p(5p + 1)(5p – 1)
= 2p[(5p)2 – (1)2]
= 2p(25p2 – 1)
= 50p3 – 2p 7

8. Find Greater Powers of Binomials
Example 5
Find each product.
(a) (7 – 4s)3
= (7 – 4s)(7 – 4s)2
= (7 – 4s)[72 – 2(7)(4s) + (4s)2]
= (7 – 4s)(49 – 56s + 16s2)
= 343 – 392s + 112s2 – 196s + 224s2 – 64s3
= 343 – 588s + 336s2 – 64s3

9. Find Greater Powers of Binomials
Example 5 (concluded)
Find each product.
(b) (2t + 3v)4
= (2t + 3v)2(2t + 3v)2
= [(2t)2 + 2(2t)(3v) + (3v)2][(2t)2 + 2(2t)(3v) + (3v)2]
= (4t2 + 12tv + 9v2)(4t2 + 12tv + 9v2)
= 16t4 + 48t3v + 36t2v2 + 48t3v + 144t2v tv3
+ 36t2v tv3 + 81v4
= 16t4 + 96t3v + 216t2v tv3 + 81v4