### Multiplying Special Products

```Objective
The student will be able to:
use patterns to multiply special
binomials.
SOL: A.2b
Designed by Skip Tyler, Varina High School
There are formulas (shortcuts) that
work for certain polynomial
multiplication problems.
2
b)
2
a
2
b
(a +
= + 2ab +
(a - b)2 = a2 – 2ab + b2
2
2
(a - b)(a + b) = a - b
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.
Let’s 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, I’ll show you both ways.
1) Multiply (x + 4)(x + 4)
First terms: x2
Outer terms: +4x
Inner terms: +4x
Last terms: +16
Combine like terms.
x2 +8x + 16
Notice you
have two of
the same
x
x
+4
x2
+4x
+4 +4x +16
Now let’s do it with the shortcut!
1) Multiply: (x + 4)2
That’s why
the 2 is in
the formula!
using (a + b)2 = a2 + 2ab + b2
a is the first term, b is the second term
(x + 4)2
a = x and b = 4
Plug into the formula
a2 + 2ab + b2
(x)2 + 2(x)(4) + (4)2
This is the
Simplify.
x2 + 8x+ 16
2) Multiply: (3x + 2y)2
using (a + b)2 = a2 + 2ab + b2
(3x + 2y)2
a = 3x and b = 2y
Plug into the formula
a2 + 2ab + b2
(3x)2 + 2(3x)(2y) + (2y)2
Simplify
9x2 + 12xy +4y2
Multiply (2a + 3)2
1.
2.
3.
4.
4a2 – 9
4a2 + 9
4a2 + 36a + 9
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.
2.
3.
4.
x2 + 2xy + y2
x2 – 2xy + y2
x2 + y2
x2 – y2
5) Multiply (x – 3)(x + 3)
First terms: x2
Outer terms: +3x
Inner terms: -3x
Last terms: -9
Combine like terms.
x2 – 9
Notice the
middle terms
eliminate
each other!
x
-3
x2
-3x
+3 +3x
-9
x
This is called the difference of squares.
5) Multiply (x – 3)(x + 3) using
2
2
(a – b)(a + b) = a – b
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
6) Multiply: (y – 2)(y + 2)
(y)2 – (2)2
y2 – 4
7) Multiply: (5a + 6b)(5a – 6b)
(5a)2 – (6b)2
25a2 – 36b2
Multiply (4m – 3n)(4m + 3n)
1.
2.
3.
4.
16m2 – 9n2
16m2 + 9n2
16m2 – 24mn - 9n2
16m2 + 24mn + 9n2
```