Lesson 83: Factoring Special Products

New Concept

The factored form of a difference of two squares is:

What’s next

Next, you'll apply this pattern and others to factor trinomials, binomials, and solve real-world problems involving area.

Property

The factored form of a perfect-square trinomial is: $a^2 + 2ab + b^2 = (a+b)^2$ and $a^2 - 2ab + b^2 = (a-b)^2$.

Explanation

Think of this as a factoring superpower! If your trinomial's first and last terms are perfect squares, check if the middle term is twice their square roots' product. If it matches, you've found a special product that factors into a neat binomial square. It is like finding a secret pattern to solve it!

Examples

$x^2 + 14x + 49 = x^2 + 2(x)(7) + 7^2 = (x+7)^2$
$36x^2 - 48x + 16 = 4(9x^2 - 12x + 4) = 4((3x)^2 - 2(3x)(2) + 2^2) = 4(3x - 2)^2$
$x^2 + 10x + 25 = x^2 + 2(x)(5) + 5^2 = (x+5)^2$

Let’s see if this trinomial hides a special pattern inside. This relates to our first key idea, factoring perfect-square trinomials.

Example Problem

Determine if $50x^2 + 40x + 8$ is a perfect-square trinomial. If so, factor it.

Step-by-Step

1. First, we examine the trinomial $50x^2 + 40x + 8$. We can see that the greatest common factor is $2$.

2. Now, let's see if the expression inside the parentheses, $25x^2 + 20x + 4$, fits the form $a^2 + 2ab + b^2$.

The first term is a perfect square: $25x^2 = (5x)^2$.
The last term is a perfect square: $4 = 2^2$.

3. We check if the middle term matches $2ab$. Let $a = 5x$ and $b=2$. Then $2ab = 2(5x)(2) = 20x$. This matches the middle term.
4. Since it fits the pattern, we can write it in factored form, remembering the GCF we factored out in step 1.

So, the polynomial is a perfect-square trinomial.

Property

The factored form of a difference of two squares is: $a^2 - b^2 = (a+b)(a-b)$.

Explanation

This is the easiest special product to spot! When you see a perfect square minus another perfect square, just take the square root of each. Your factors are simply the sum of those roots and the difference of those roots. No tricky middle term to worry about, just pure subtraction magic!

Examples

$4x^2 - 25 = (2x)^2 - 5^2 = (2x+5)(2x-5)$
$9m^4 - 16n^6 = (3m^2)^2 - (4n^3)^2 = (3m^2 + 4n^3)(3m^2 - 4n^3)$
$-81 + x^{10} = x^{10} - 81 = (x^5)^2 - 9^2 = (x^5+9)(x^5-9)$

Now let’s look for a different pattern, the difference of two squares. This example highlights our second key idea for today's lesson.

Example Problem

Determine if $25x^8 - 49y^2$ is a difference of two squares. If so, factor it.

Step-by-Step

1. We start with the expression $25x^8 - 49y^2$. We need to check if both terms are perfect squares.
2. We factor each term to see if it can be written as a square.

3. We can now rewrite the expression as a difference of two squares, in the form $a^2 - b^2$.

4. With $a = 5x^4$ and $b = 7y$, we can apply the difference of two squares formula, $a^2 - b^2 = (a+b)(a-b)$.

Property

Real-world problems involving area can often be simplified by factoring special products.

Explanation

Factoring isn't just for math class! It can help you figure out changes in area, like how much a radio signal's reach has grown, or the area of a border around a pool. It turns complicated-looking polynomials into simple, meaningful measurements that you can actually use in real life situations!

Examples

A signal area grows to $\pi(r^2 + 12r + 36)$. Factoring gives $\pi(r+6)^2$, so the radius increased by 6 miles.
A square deck with side $x$ has an 8-foot square shed. The paintable area is $x^2 - 64$, which factors to $(x-8)(x+8)$ square feet.
The cost difference between two projects is $c^2 - 900$ dollars. Factoring gives $(c-30)(c+30)$ dollars.