Multiply Binomials with Patterns

1
ObjectiveThe student will be able to
use patterns to multiply special binomials. SOL
A.2b
Designed by Skip Tyler, Varina High School
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.

3
Lets try one!1) Multiply (x 4)2

• You can multiply this by rewriting this as (x
  4)(x 4)
• OR
• You can use the following rule as a shortcut
• (a b)2 a2 2ab b2
• For comparison, Ill show you both ways.

4
1) Multiply (x 4)(x 4)
Notice you have two of the same answer?
x2

• First terms
• Outer terms
• Inner terms
• Last terms
• Combine like terms.
• x2 8x 16

x 4
x
4
4x
4x
x2
4x
16
4x
16
Now lets do it with the shortcut!
5
1) Multiply (x 4)2 using (a b)2 a2 2ab
b2
Thats why the 2 is in the formula!

• a is the first term, b is the second term
• (x 4)2
• a x and b 4
• Plug into the formulaa2 2ab b2
• (x)2 2(x)(4) (4)2
• Simplify.
• x2 8x 16

This is the same answer!
6
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)2
• Simplify
• 9x2 12xy 4y2

7
Multiply (2a 3)2

1. 4a2 9
2. 4a2 9
3. 4a2 36a 9
4. 4a2 12a 9

8
Multiply (x 5)2 using (a b)2 a2 2ab
b2Everything 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

9
Multiply (x y)2

1. x2 2xy y2
2. x2 2xy y2
3. x2 y2
4. x2 y2

10
5) Multiply (x 3)(x 3)
Notice the middle terms eliminate each other!
x2

• First terms
• Outer terms
• Inner terms
• Last terms
• Combine like terms.
• x2 9

x -3
x
3
3x
-3x
x2
-3x
-9
3x
-9
This is called the difference of squares.
11
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

12
6) Multiply (y 2)(y 2)

• (y)2 (2)2
• y2 4
• 7) Multiply (5a 6b)(5a 6b)
• (5a)2 (6b)2
• 25a2 36b2

13
Multiply (4m 3n)(4m 3n)

1. 16m2 9n2
2. 16m2 9n2
3. 16m2 24mn - 9n2
4. 16m2 24mn 9n2