Lesson Objectives
• Demonstrate an understanding of special polynomial products
• Learn how to factor a difference of squares
• Learn how to factor a perfect square trinomial
• Learn how to factor a difference of cubes
• Learn how to factor a sum of cubes

Special Factoring

When we learned how to multiply polynomials, we learned how to quickly multiply commonly occurring scenarios using "special products" formulas. When we reverse these formulas, we end up with the factored form, this is referred to as "special factoring".

The Difference of Two Squares

When we see the difference of two squares, we have a binomial (two-term polynomial) where each term is squared. The second term should be subtracted away from the first term.
x2 - y2 = (x + y)(x - y)
Let's look at an example.
Example 1: Factor each.
18x2 - 32
First and foremost, we can factor out a common factor of 2:
2(9x2 - 16)
To match our formula, we can rewrite this as:
2((3x)2 - (4)2)
Following our formula, this can be factored as:
2(3x + 4)(3x - 4)

Perfect Square Trinomials

A perfect square trinomial is one that factors into a binomial squared. We have two different scenarios, one with a sum and another with a difference.
x2 + 2xy + y2 = (x + y)2
x2 - 2xy + y2 = (x - y)2
Let's look at a few examples.
Example 2: Factor each.
125x2 + 50x + 5
First and foremost, we can factor out a common factor of 5:
5(25x2 + 10x + 1)
To match our formula, we can rewrite this as:
5((5x)2 + 2 • 5x • 1 + (1)2)
Following our formula, this can be factored as:
5(5x + 1)2
Example 3: Factor each.
12x2 - 36xy + 27y2
First and foremost, we can factor out a common factor of 3:
3(4x2 - 12xy + 9y2)
To match our formula, we can rewrite this as:
3((2x)2 - 2 • 2x • 3y + (3y)2)
Following our formula, this can be factored as:
3(2x - 3y)2

The Sum of Cubes

When we have the sum of two cubes, we can use the following formula to find the factorization:
x3 + y3 = (x + y)(x2 - xy + y2)
Let's look at an example.
Example 4: Factor each.
500x3 + 4
First and foremost, we can factor out a common factor of 4:
4(125x3 + 1)
To match our formula, we can rewrite this as:
4((5x)3 + (1)3)
Following our formula, this can be factored as:
4(5x + 1)((5x)2 - 5x • 1 + (1)2)
4(5x + 1)(25x2 - 5x + 1)

The Difference of Cubes

When we have the difference of cubes, we can use the following formula to find the factorization:
x3 - y3 = (x - y)(x2 + xy + y2)
Let's look at an example.
Example 5: Factor each.
108x3 - 32y3
First and foremost, we can factor out a common factor of 4:
4(27x3 - 8y3)
To match our formula, we can rewrite this as:
4((3x)3 - (2y)3)
Following our formula, this can be factored as:
4(3x - 2y)((3x)2 + (3x)(2y) + (2y)2)
4(3x - 2y)(9x2 + 6xy + 4y2)

Skills Check:

Example #1

Factor each. $$5x^2 - 45$$

A
$$25(x - 3)^2$$
B
$$5(x + 3)(x - 3)$$
C
$$5(x + 9)^2$$
D
$$5(x + 5)(x - 5)$$
E
$$5(x - 3)^2$$

Example #2

Factor each. $$100x^2 - 160xy + 64y^2$$

A
$$4(5x - 4y)^2$$
B
$$16(5x + 4y)(5x - 4y)$$
C
$$(5x + 4y)(5x - 4y)$$
D
$$4(25x + 16y)^2$$
E
$$2(5x - 2y)^2$$

Example #3

Factor each. $$64 - 27x^3$$

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A
$$(3x - 9)(x + 1)(x - 3)$$
B
$$(3x + 4)^3$$
C
$$(3x + 4)(9x^2 + 12x - 16)$$
D
$$-(3x - 4)(9x^2 + 12x + 16)$$
E
$$(-3x - 4)^3$$

Example #4

Factor each. $$x^2 - 4x + 4 - y^4$$

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A
Prime
B
$$(x + 2 + y^2)(y^2 - x + 2)$$
C
$$-(x - 2 + y^2)(y^2 - x + 2)$$
D
$$(x + 2y)(x - y)^2$$
E
$$2(x + 4y)^3$$