9.4

Special Cases

9.4 – Special Cases

Goals / “I can…”I can find the square of a binomialI can find the difference of squares

9.4 – Special Cases

IMPORTANT: If you don’t want to take the rules and store them in your brain to recall at any time, you can always FOIL the binomials like yesterday.

9.4 – Special Cases

There are certain binomial combinations that occur frequently. They form patterns when multiplied. If you remember the pattern, you can quickly simplify.

9.4 – Special Cases

Binomial Square:

(x + 2)Most people get this wrong because

they distribute the square and get

x + 4x + 4However, it really means

(x + 2)(x + 2)so you have to FOIL it.

2

2

Square of a Binomial

Square the first term:

2nm

2m 2mn

2n22 2 nmnm

• Square the last term:

• Double the product of both terms:

2) Multiply: (3x + 2y)2

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

(3x + 2y)2

a = 3x and b = 2y

Plug into the formulaa2 + 2ab + b2

(3x)2 + 2(3x)(2y) + (2y)2Simplify

9x2 + 12xy +4y2

Example #1

243 ba

29a ab24 216b

Example #2

22 sr

24r rs4 2s

Multiply (2a + 3)2

1. 4a2 – 9

2. 4a2 + 9

3. 4a2 + 36a + 9

4. 4a2 + 12a + 9

Multiply: (x – 5)2

using (a – b)2 = a2 – 2ab + b2

Everything is the same except the signs!

(x)2 – 2(x)(5) + (5)2

x2 – 10x + 25

4) Multiply: (4x – y)2

(4x)2 – 2(4x)(y) + (y)2

16x2 – 8xy + y2

Multiply (x – y)2

1. x2 + 2xy + y2

2. x2 – 2xy + y2

3. x2 + y2

4. x2 – y2

9.4 – Special Cases

Difference of Squares:Difference of Squares:

(2x + 3)(2x – 3)

notice they are the same except for the sign

Product of the Sum & Difference

Square the first term:

nmnm 2m

• Square the last term:2n

• Write the difference of the two squares:

22 nm

9.4 – Special Cases

(2x + 3)(2x – 3) =

4x – 6x + 6x – 9 =

4x – 9

2

2

5) Multiply (x – 3)(x + 3) using (a – b)(a + b) = a2 – b2

You can only use this rule when the binomials are exactly the same except for

the sign.

(x – 3)(x + 3)

a = x and b = 3

(x)2 – (3)2

x2 – 9

Example #2

25

55 zz

2z

Multiply (4m – 3n)(4m + 3n)

1. 16m2 – 9n2

2. 16m2 + 9n2

3. 16m2 – 24mn - 9n2

4. 16m2 + 24mn + 9n2

9.4 – Special Cases

(x + 7)2

9.4 – Special Cases

(x – 9)2

There are formulas (shortcuts) that work for certain polynomial

multiplication problems.

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

(a - b)2 = a2 – 2ab + b2

(a - b)(a + b) = a2 - b2

Being able to use these formulas will help you in the future when you have to factor. If you do not remember the formulas, you can always multiply

using distributive, FOIL, or the box method.