factoring polynomials

Here is a step-by-step breakdown of how to solve every type of factoring problem shown across the three PowerPoint slide decks in your notes.

1. Common Factors & Grouping (Lessons 06.01 & 06.02)

A. Factoring Out the Greatest Common Factor (GCF)

Look at the numbers (coefficients) and variables across every term to find the biggest common factor.

Find the GCF of the numbers: Look for the highest number that divides evenly into all coefficients.
Find the GCF of the variables: Take the smallest exponent of any shared variable.
Divide each term by that GCF and write the remaining terms inside brackets.

Example: $12x - 8y + 16z$
- GCF of $12, 8,$ and $16$ is $4$ .
- Divide each term by $4$ : $12x/4 = 3x$ , $-8y/4 = -2y$ , $16z/4 = 4z$ .
- Answer: $4(3x - 2y + 4z)$
Example with exponents: $18ab^2 - 9a^2b$
- GCF of numbers is $9$ .
- GCF of variables: lowest power of $a$ is $a^1$ , lowest power of $b$ is $b^1$ . GCF = $9ab$ .
- Answer: $9ab(2b - a)$

B. Factoring Formulas & Binomial GCFs

When a common binomial factor exists in multiple terms, pull out the entire bracket as the GCF.

Formula Example: $SA = 2\pi r^2 + 2\pi r h$
- Both terms share $2\pi r$ .
- Answer: $SA = 2\pi r(r + h)$
Grouping Example: $8p^2 - 4p - 10p + 5$
- Group the first two and last two terms: $(8p^2 - 4p) - (10p - 5)$ .
- Factor out GCF from each pair: $4p(2p - 1) - 5(2p - 1)$ .
- Pull out the shared $(2p - 1)$ : $(4p - 5)(2p - 1)$

2. Factoring Trinomials ( $x^2 + bx + c$ ) (Lessons 06.03 & 06.04)

When factoring a simple quadratic trinomial where the leading coefficient is $1$ :

Find two numbers that multiply to give $c$ (the last term) and add up to $b$ (the middle coefficient).
Write the factored form as $(x + \text{first number})(x + \text{second number})$ .

Cases to Watch For:

Both Middle & Last Terms Positive: $x^2 + 11x + 10$
- Multiply to $+10$ , add to $+11$ $\rightarrow$ $+1$ and $+10$ .
- Answer: $(x + 1)(x + 10)$
Middle Negative, Last Positive: $x^2 - 3x + 2$
- Both numbers must be negative to multiply to a positive and add to a negative.
- Multiply to $+2$ , add to $-3$ $\rightarrow$ $-1$ and $-2$ .
- Answer: $(x - 1)(x - 2)$
Last Term Negative: $x^2 - 2x - 15$
- One number must be positive and one negative.
- Multiply to $-15$ , add to $-2$ $\rightarrow$ $-5$ and $+3$ .
- Answer: $(x - 5)(x + 3)$
Not Possible / Prime: $x^2 - 12x + 24$
- Factors of $24$ : $(1, 24), (2, 12), (3, 8), (4, 6)$ . None of these add or subtract to $-12$ .
- Answer: NOT POSSIBLE

💡 Pro-Tip: Always look for a GCF first before factoring the trinomial!

Example: $2n^2 + 14n + 24$

Pull out $2$ first: $2(n^2 + 7n + 12)$ .

Factor the inside ( $+3$ and $+4$ multiply to $12$ and add to $7$ ).

Answer: $2(n + 3)(n + 4)$

3. Special Cases: Difference of Squares & Multi-Step Factoring (Lessons 06.05 & 06.06)

A. Difference of Squares ( $a^2 - b^2$ )

A binomial is a Difference of Squares if:

It is a subtraction problem (minus sign).
Both terms are perfect squares.

Use the pattern: $a^2 - b^2 = (a - b)(a + b)$

Example: $x^2 - 36$
- Square root of $x^2$ is $x$ ; square root of $36$ is $6$ .
- Answer: $(x - 6)(x + 6)$
Example: $4x^2 - 9y^2$
- $\sqrt{4x^2} = 2x$ , $\sqrt{9y^2} = 3y$ .
- Answer: $(2x + 3y)(2x - 3y)$
Sum of Squares: $4x^2 + 49$
- Addition between two squares cannot be factored over real numbers.
- Answer: NOT POSSIBLE

B. Multi-Step / Fully Factorizing

Sometimes you have to factor a problem more than once!

Combine GCF + Difference of Squares: $3x^2 - 12y^2$
1. Pull out GCF of $3$ : $3(x^2 - 4y^2)$ .
2. Factor $(x^2 - 4y^2)$ as a difference of squares.
3. Answer: $3(x - 2y)(x + 2y)$
Double Difference of Squares: $x^4 - 16$
1. First pass: $(x^2 - 4)(x^2 + 4)$ .
2. $(x^2 - 4)$ can be factored again! ( $(x^2 + 4)$ is a sum, so leave it alone).
3. Answer: $(x + 2)(x - 2)(x^2 + 4)$