7.4 Factor Special Products

Factoring special products is about recognizing specific patterns in polynomials and applying formulas to break them into simpler factors. Once you can spot these patterns quickly, factoring becomes much faster and more reliable than trial and error.

Factoring Special Products

Factoring perfect square trinomials

A perfect square trinomial is what you get when you square a binomial. Recognizing one lets you factor it instantly instead of guessing.

The two forms are:

• $a^2 + 2ab + b^2 = (a + b)^2$
• $a^2 - 2ab + b^2 = (a - b)^2$

To check whether a trinomial is a perfect square:

1. Confirm the first and last terms are both perfect squares.
2. Take the square roots of those two terms.
3. Multiply those square roots together and double the result.
4. If that matches the middle term (ignoring sign), you have a perfect square trinomial.
5. The sign of the middle term tells you whether the binomial uses $+$ or $-$.

Example: Factor $x^2 + 6x + 9$

• $x^2$ is a perfect square ($a = x$), and $9$ is a perfect square ($b = 3$).
• Twice the product of the square roots: $2(x)(3) = 6x$. That matches the middle term.
• So $x^2 + 6x + 9 = (x + 3)^2$.

Example: Factor $9x^2 - 12x + 4$

• $9x^2$ is a perfect square ($a = 3x$), and $4$ is a perfect square ($b = 2$).
• Twice the product: $2(3x)(2) = 12x$. The middle term is $-12x$, so the sign is negative.
• So $9x^2 - 12x + 4 = (3x - 2)^2$.

Sums and differences of cubes

These formulas let you factor expressions where two cubed terms are added or subtracted. They're worth memorizing because you can't factor cubes by the same methods you use for squares.

• Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

A helpful way to remember the signs: the first factor matches the sign in the original expression. In the second factor (the trinomial), the first term is always positive, the middle term has the opposite sign of the original, and the last term is always positive.

Example: Factor $x^3 + 8$

1. Identify the cube roots: $x^3 \to a = x$ and $8 \to b = 2$ (since $2^3 = 8$).

2. Write the first factor with the same sign: $(x + 2)$.

3. Build the trinomial: $x^2 - 2x + 4$.

4. Result: $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$.

Example: Factor $64m^3 - 125$

1. Cube roots: $64m^3 \to a = 4m$ and $125 \to b = 5$.

2. First factor (same sign as original): $(4m - 5)$.

3. Trinomial (opposite, always positive): $(4m)^2 + (4m)(5) + 5^2 = 16m^2 + 20m + 25$.

4. Result: $64m^3 - 125 = (4m - 5)(16m^2 + 20m + 25)$.

Recognition of special product patterns

Before you can apply a formula, you need to identify which pattern you're looking at. Here's a quick reference:

Methods for complete polynomial factoring

When you're given a polynomial to factor completely, follow these steps in order:

1. Factor out the GCF first. Always look for a greatest common factor before anything else. This simplifies what's left and can reveal a special pattern hiding underneath.

2. Count the terms and look for patterns.

   • Two terms: check for difference of squares, sum of cubes, or difference of cubes.
   • Three terms: check for a perfect square trinomial; if not, use trial and error or the ac method.
   • Four terms: try factoring by grouping.

3. Factor each resulting factor again. After your first round of factoring, check whether any factor can be broken down further.

Example: Factor $2x^3 + 6x^2 - 8x$ completely.

1. GCF is $2x$: $2x^3 + 6x^2 - 8x = 2x(x^2 + 3x - 4)$

2. The trinomial $x^2 + 3x - 4$ isn't a perfect square, so factor it normally. You need two numbers that multiply to $-4$ and add to $3$: that's $4$ and $-1$.

3. Result: $2x(x + 4)(x - 1)$

Example: Factor $3x^3 - 75x$ completely.

1. GCF is $3x$: $3x^3 - 75x = 3x(x^2 - 25)$

2. $x^2 - 25$ is a difference of squares: $(x + 5)(x - 5)$

3. Result: $3x(x + 5)(x - 5)$

The key habit is to always pull out the GCF first. Many students skip this step and then struggle to spot the special pattern that's buried inside.