Special Products

1
Special Products of Polynomials
2
Objectives

1. Recognize special polynomial product patterns.
2. Use special polynomial product patterns to
   multiply two polynomials.

3
Review
Multiplication of polynomials is an extension of
the distributive property. When you multiply two
polynomials you distribute each term of one
polynomial to each term of the other polynomial.
We can multiply polynomials in a vertical format
like we would multiply two numbers.
(x 3)
(x 2)
x
_________
6
2x
0
3x
x2
_________
x2 5x 6
4
Review
Multiplication of polynomials is an application
of the distributive property. When you multiply
two polynomials you distribute each term of one
polynomial to each term of the other polynomial.
We can also multiply polynomials by using the
FOIL pattern.
(x 3)(x 2)
x2 5x 6
x(x)
x(2)
(3)(x)
(3)(2)
5
Special Products
Some pairs of binomials have special products.
When multiplied, these pairs of binomials always
follow the same pattern.
By learning to recognize these pairs of
binomials, you can use their multiplication
patterns to find the product quicker and easier.
6
Special Products
One special pair of binomials is the sum of two
numbers times the difference of the same two
numbers.
Lets look at the numbers x and 4. The sum of x
and 4 can be written (x 4). The difference of
x and 4 can be written (x 4).
Their product is
(x 4)(x 4)
x2 4x 4x 16
x2 16
Multiply using foil, then collect like terms.
7
Special Products
Here are more examples

(x 4)(x 4)
x2 4x 4x 16
x2 16
What do all of these have in common?
(x 3)(x 3)
x2 3x 3x 9
x2 9
(5 y)(5 y)
25 5y 5y y2
25 y2
8
Special Products
What do all of these have in common?
x2 16
x2 9
25 y2
They are all binomials.
They are all differences.
Both terms are perfect squares.
9
Special Products
For any two numbers a and b, (a b)(a b) a2
b2.
In other words, the sum of two numbers times the
difference of those two numbers will always be
the difference of the squares of the two numbers.
Example
(x 10)(x 10) x2 100
(5 2)(5 2) 25 4 21
3 7 21
10
Special Products
The other special products are formed by squaring
a binomial.
(x 4)2 and (x 6)2 are two example of
binomials that have been squared.
Lets look at the first example (x 4)2
(x 4)2 (x 4)(x 4)
x2
4x
16
4x
x2 8x 16
Now we FOIL and collect like terms.
11
Special Products
Whenever we square a binomial like this, the same
pattern always occurs.
(x 4)2 (x 4)(x 4)
x2
4x
16
4x
x2 8x 16
In the final product it is squared
See the first term?
and it appears in the middle term.
12
Special Products
Whenever we square a binomial like this, the same
pattern always occurs.
(x 4)2 (x 4)(x 4)
x2
4x
16
4x
x2 8x 16
What about the second term?
The middle number is 2 times 4
and the last term is 4 squared.
13
Special Products
Whenever we square a binomial like this, the same
pattern always occurs.
(x 4)2 (x 4)(x 4)
x2
4x
16
4x
x2 8x 16
Squaring a binomial will always produce a
trinomial whose first and last terms are perfect
squares and whose middle term is 2 times the
numbers in the binomial, or
For two numbers a and b, (a b)2 a2 2ab b2
14
Special Products
Is it the same pattern if we are subtracting, as
in the expression (y 6)2?
(y 6)2 (y 6)(y 6)
y2
6y
36
6y
y2 12y 36
It is almost the same. The y is squared, the 6
is squared and the middle term is 2 times 6 times
y. However, in this product the middle term is
subtracted. This is because we were subtracting
in the original binomial. Therefore our rule has
only one small change when we subtract.
For any two numbers a and b, (a b)2 a2 2ab
b2
15
Special Products
Examples
(x 3)2
(x 3)(x 3)
Remember (a b)2 a2 2ab b2
x2 2(3)(x) 32
x2 6x 9
(z 4)2
Remember (a b)2 a2 2ab b2
(z 4)(z 4)
z2 2(4)(z) 42
z2 8z 16
16
Special Products
You should copy these rules into your notes and
try to remember them. They will help you work
faster and make many problems you solve easier.
For any two numbers a and b, (a b)(a b) a2
b2.
For two numbers a and b, (a b)2 a2 2ab b2
For any two numbers a and b, (a b)2 a2 2ab
b2
17
You Try It.

1. (2x 5)(2x 5)
2. (x 7)2
3. (x 2)2
4. (2x 3y)2

18
You Try It.

1. (2x 5)(2x 5)

(2x 5)(2x 5)
22x2 52
4x2 25
19
You Try It.
2. (x 7)2
(x 7)2
x2 2(7)(x) 72
x2 14x 49
20
You Try It.
3. (x 2)2
(x 2)2
x2 2(2)(x) 22
x2 4x 4
21
You Try It.
4. (2x 3y)2
(2x 3y)2
22x2 2(2x)(3y) 32y2
4x2 12x 9y2