5.6

Special Products

When Squaring Binomials… Remember:

2 2 2.2x y x xy y

2 2 2.2x y x xy y

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

2 2 2 2 2, not2 x y x xy y x y

EXAMPLE 1Find (x + 4)2.

Solution: 4 4x x 2 4 4 16x x x 2 8 16x x

EXAMPLE 2Square each binomial and simplify.

Solution: 22 1x

25 6r s

22 7x x

24 2 2 1x x x 2 1 2 1x x 24 4 1x x

5 6 5 6r s r s 2 225 30 30 36r rs rs s 2 225 60 36r rs s

2 7 2 7x x x 24 14 14 49x x x x

24 28 49x x x 3 24 28 49x x x

Find the product of the sum and difference of two terms.

In binomial products of the form (x + y)(x − y), one binomial is the sum of two terms and the other is the difference of the same two terms. Consider (x + 2)(x − 2).

22 2 2 2 4x x x x x

Thus, the product of x + y and x − y is the difference of two squares.

2 2– –x + y x y x y

2 4x

CONJUGATE PAIRS

EXAMPLE 3Find the product.

Solution:

3 3y y

2 23 y

29 y

EXAMPLE 4 Find each product.

10 7 10 7m m

6 5 6 5x x x

2100 49m 2 210 7m

2 26 5x x 236 25x x

336 25x x

34 1x 24 1 4 1x x

2 23 2 3 2k k 43 2k

216 8 1 4 1x x x 3 2 264 32 4 16 8 1x x x x x 3 264 48 12 1x x x

2 29 12 4 9 12 4k k k k

4 3 2 3 2 281 108 36 108 144 48 36 48 16k k k k k k k k 4 3 281 216 216 96 16k k k k

Homework5.1: 1 – 87 EOO5.2: 1 – 77 ODD5.3: 1 – 83 EOO5.4 1 – 95 EOO5.5 1 – 83 EOO5.6 3 – 53 ODD 71 – 73 ODD