Factoring Polynomials

Understanding Polynomials Before We Factor

Before we factor, we need to understand what a polynomial is. A polynomial is an expression made up of terms added or subtracted together. Each term has a coefficient (number) and variables with exponents.

Example: 3x² + 7x - 5

First term: 3x² (coefficient is 3)
Second term: 7x (coefficient is 7)
Third term: -5 (this is a constant)

When we factor, we're finding what we can pull out of all terms. Think of it like finding common ingredients in a recipe.

Method 1: Greatest Common Factor (GCF)

What is GCF?

The Greatest Common Factor is the largest number or variable that divides evenly into all terms. It's like finding the biggest piece that all terms share.

Understanding GCF

Look at 12x² + 8x

What divides into 12? 1, 2, 3, 4, 6, 12
What divides into 8? 1, 2, 4, 8
What divides into both? 1, 2, 4 → Greatest is 4

Both terms also have at least one x, so our GCF is 4x.

Steps for GCF Factoring

Find the GCF of all coefficients
Find the lowest power of each variable that appears in every term
Factor out the GCF
Write what's left in parentheses

Detailed Example

Factor 6x³ + 9x² - 12x

Step 1: GCF of numbers (6, 9, 12)

6 = 2 × 3
9 = 3 × 3
12 = 2 × 2 × 3
Common factor: 3

Step 2: Each term has at least one x → x¹

Step 3: GCF = 3x

Step 4: 6x³ + 9x² - 12x = 3x(2x² + 3x - 4)

Practice Problem

Factor each polynomial using GCF:

5a + 10b
4x² + 8x
15m³ + 10m² - 5m

Method 2: Difference of Two Squares (DOTS)

What is DOTS?

DOTS stands for Difference Of Two Squares. This is when you have two perfect squares being subtracted.

The pattern: a² - b² = (a + b)(a - b)

Recognizing Perfect Squares

A perfect square is a number or expression that can be written as something times itself.

Perfect Square Recognition

x² = x × x → perfect square
4 = 2 × 2 = 2² → perfect square
9x² = 3x × 3x = (3x)² → perfect square
x = x¹ → NOT a perfect square
3x² → NOT a perfect square (3 isn't a perfect square)

How to Factor Using DOTS

Confirm both terms are perfect squares
Confirm there's a minus sign between them
Take the square root of each term
Write as (square root 1 + square root 2)(square root 1 - square root 2)

Detailed Example

Factor x² - 25

Step 1: Is x² a perfect square? Yes, square root is x

Step 2: Is 25 a perfect square? Yes, square root is 5

Step 3: Do we have a minus sign? Yes ✓

Step 4: x² - 25 = (x + 5)(x - 5)

More Complex Example

Factor 9a⁴ - 16b²

Step 1: Square root of 9a⁴ = 3a²

Step 2: Square root of 16b² = 4b

Step 3: Both are perfect squares with minus sign ✓

Step 4: 9a⁴ - 16b² = (3a² + 4b)(3a² - 4b)

Practice Problem

Factor each polynomial using DOTS:

x² - 16
4a² - 9
m² - 100n²

Method 3: Sum and Difference of Cubes (SDOTC)

What is SDOTC?

SDOTC handles Sum and Difference of cubes. These are similar to DOTS but with cubes instead of squares.

Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)

Recognizing Perfect Cubes

A perfect cube is something multiplied by itself three times.

Perfect Cube Recognition

x³ = x × x × x → perfect cube
8 = 2 × 2 × 2 = 2³ → perfect cube
27a³ = 3a × 3a × 3a = (3a)³ → perfect cube
64b⁶ = 4b² × 4b² × 4b² = (4b²)³ → perfect cube

How to Factor Using Sum of Cubes

For a³ + b³:

Find the cube roots: find a and b
Write the first binomial: (a + b)
For the trinomial: (a² - ab + b²)
Answer: (a + b)(a² - ab + b²)

Sum of Cubes Example

Factor x³ + 8

Step 1: Cube root of x³ = x and cube root of 8 = 2

Step 2: First binomial: (x + 2)

Step 3: Trinomial: (x² - 2x + 4)

a² = x²
ab = x × 2 = 2x (we put minus here)
b² = 2² = 4

Step 4: x³ + 8 = (x + 2)(x² - 2x + 4)

How to Factor Using Difference of Cubes

For a³ - b³:

Find the cube roots: find a and b
Write the first binomial: (a - b)
For the trinomial: (a² + ab + b²) ← Note: plus signs here
Answer: (a - b)(a² + ab + b²)

Difference of Cubes Example

Factor 27m³ - 64

Step 1: Cube root of 27m³ = 3m and cube root of 64 = 4

Step 2: First binomial: (3m - 4)

Step 3: Trinomial: (9m² + 12m + 16)

a² = (3m)² = 9m²
ab = 3m × 4 = 12m (plus here!)
b² = 4² = 16

Step 4: 27m³ - 64 = (3m - 4)(9m² + 12m + 16)

Practice Problem

Factor each polynomial using SDOTC:

a³ + 27
b³ - 125
8x³ + 1

Method 4: Perfect Square Trinomials (PST)

What is a Perfect Square Trinomial?

A perfect square trinomial is when a binomial is squared. It follows a pattern:

(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²

When factoring, we reverse this process to find the original binomial.

Recognizing PST

Look for:

First term is a perfect square
Last term is a perfect square
Middle term equals 2 × square root of first × square root of last

Recognizing PST

Is x² + 6x + 9 a perfect square trinomial?

First term: x² → square root is x ✓
Last term: 9 → square root is 3 ✓
Middle check: 2 × x × 3 = 6x → matches! ✓

This IS a perfect square trinomial!

How to Factor PST

Take the square root of the first term
Take the square root of the last term
Determine the sign from the middle term
Write as (a ± b)²

PST Factoring Examples

Example 1: x² + 8x + 16

Square root of x² = x
Square root of 16 = 4
Middle term is positive: +8x
Answer: (x + 4)²

Example 2: 4a² - 12a + 9

Square root of 4a² = 2a
Square root of 9 = 3
Middle term is negative: -12a
Check: 2 × 2a × 3 = 12a ✓
Answer: (2a - 3)²

Practice Problem

Factor each polynomial if it's a perfect square trinomial:

m² + 10m + 25
4b² - 4b + 1
x² + 5x + 4 (Is this PST?)

Method 5: Quadratic Trinomials (QT)

What are Quadratic Trinomials?

A quadratic trinomial is a polynomial with three terms in the form ax² + bx + c where a, b, and c are numbers.

When we can't use the other methods, we use the quadratic trinomial method.

The AC Method (Most Reliable)

Step 1: Multiply a × c

Step 2: Find two numbers that multiply to give ac and add to give b

Step 3: Rewrite the middle term using these two numbers

Step 4: Factor by grouping

AC Method - Simple Example

Factor x² + 5x + 6

Step 1: a = 1, c = 6 → a × c = 6

Step 2: We need two numbers that:

Multiply to give 6: (1,6), (2,3)
Add to give 5 (the b value): 2 + 3 = 5 ✓

Step 3: Rewrite: x² + 2x + 3x + 6

Step 4: Factor by grouping:

Group 1: x² + 2x = x(x + 2)
Group 2: 3x + 6 = 3(x + 2)
Combined: (x + 2)(x + 3)

AC Method - More Complex

Factor 2x² + 7x + 3

Step 1: a = 2, c = 3 → a × c = 6

Step 2: Two numbers that multiply to 6 and add to 7:

Pairs: (1,6), (2,3)
1 + 6 = 7 ✓

Step 3: Rewrite: 2x² + 1x + 6x + 3

Step 4: Factor by grouping:

Group 1: 2x² + 1x = x(2x + 1)
Group 2: 6x + 3 = 3(2x + 1)
Combined: (2x + 1)(x + 3)

Check: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3 ✓

When the Leading Coefficient is 1

When a = 1 (so we have x² + bx + c), we can use a simpler method:

Find two numbers that multiply to c and add to b.

Simple Trinomial Factoring

Factor x² - 7x + 10

We need two numbers that:

Multiply to 10: (1,10), (2,5)
Add to -7: -2 + (-5) = -7 ✓

Answer: (x - 2)(x - 5)

Practice Problem

Factor each quadratic trinomial:

x² + 7x + 12
2x² + 11x + 5
x² - 6x + 8

Putting It All Together: Factoring Strategy

When you see a polynomial to factor, follow these steps:

Always check for GCF first! If there's a common factor, pull it out.
Count the terms:
- 2 terms? Check for DOTS or SDOTC
- 3 terms? Check for PST or use QT method
- 4+ terms? Try factoring by grouping
After factoring, check if you can factor again. Sometimes a factor can be factored further!

Complete Factoring Example

Factor completely: 2x³ - 50x

Step 1: GCF → 2x(x² - 25)

Step 2: We have (x² - 25) with 2 terms. Is this DOTS?

x² is a perfect square ✓
25 is a perfect square ✓
Minus sign ✓

Step 3: Factor DOTS: (x - 5)(x + 5)

Final Answer: 2x(x - 5)(x + 5)

Practice Problem

Factor each polynomial completely:

3a² - 27
x³ + 6x² + 9x
4b⁴ - 16b²

Factoring Polynomials Reviewer

Understanding Polynomials Before We Factor

Method 1: Greatest Common Factor (GCF)

What is GCF?

Understanding GCF

Steps for GCF Factoring

Detailed Example

Practice Problem

Method 2: Difference of Two Squares (DOTS)

What is DOTS?

Recognizing Perfect Squares

Perfect Square Recognition

How to Factor Using DOTS

Detailed Example

More Complex Example

Practice Problem

Method 3: Sum and Difference of Cubes (SDOTC)

What is SDOTC?

Recognizing Perfect Cubes

Perfect Cube Recognition

How to Factor Using Sum of Cubes

Sum of Cubes Example

How to Factor Using Difference of Cubes

Difference of Cubes Example

Practice Problem

Method 4: Perfect Square Trinomials (PST)

What is a Perfect Square Trinomial?

Recognizing PST

Recognizing PST

How to Factor PST

PST Factoring Examples

Practice Problem

Method 5: Quadratic Trinomials (QT)

What are Quadratic Trinomials?

The AC Method (Most Reliable)

AC Method - Simple Example

AC Method - More Complex

When the Leading Coefficient is 1

Simple Trinomial Factoring

Practice Problem

Putting It All Together: Factoring Strategy

Complete Factoring Example

Practice Problem

Practice Review Problems

Answer Key