Factoring Polynomials: Grouping, Difference of Squares, and Sum/Difference of Cubes

Hello! Welcome to your first lesson. I'm excited to guide you on your journey towards mastering mathematics. Your goal of reaching "Einstein level" mathematics is ambitious and inspiring, and like any great construction project, it starts with a solid foundation. This course is designed to build that foundation, one crucial concept at a time.

Today, we'll begin with a fundamental skill in algebra: factoring polynomials. Think of factoring as breaking down a complex mathematical structure into its simplest building blocks. Just as an engineer needs to understand the properties of individual bricks and beams to build a skyscraper, you'll need to master factoring to construct and deconstruct the complex arguments of higher mathematics.

This lesson will cover three key techniques:

Factoring by grouping
Factoring the difference of squares
Factoring the sum and difference of cubes

By the end of this lesson, you will be able to recognize and apply these methods to break down polynomials into simpler expressions. Let's begin.

1. What is Factoring? The Greatest Common Factor (GCF)

Before we dive into the specific techniques, let's establish what factoring is. You're already familiar with factoring integers. For example, the number 12 can be factored into $2 \times 6$ or $3 \times 4$ . Its complete prime factorization is $2 \times 2 \times 3$ . Factoring polynomials works on the same principle: we find the simpler polynomials that multiply together to give the original polynomial.

The first step in any factoring problem should always be to look for a Greatest Common Factor (GCF). This is the largest expression (including numbers and variables) that divides into every term of the polynomial.

Algebra - Factoring Polynomials

To start, let's read a clear introduction to factoring and the Greatest Common Factor from Paul's Online Math Notes. This will solidify the core idea.

Please read the first two parts of the page: the main introduction under 'Section 1.5 : Factoring Polynomials' and the section titled 'Greatest Common Factor'. Pay close attention to Example 1, especially part (c), which highlights a common mistake.

As you saw in the reading, factoring out the GCF is like using the distributive property, $a (b + c) = ab + a c$ , in reverse. Finding a common factor in all terms allows you to "pull it out" to the front.

2. Factoring by Grouping

What happens when not all terms share a common factor? For polynomials with four terms, we can often use a technique called factoring by grouping. The strategy is to group the polynomial into two pairs of terms, factor the GCF from each pair, and then factor out a common binomial factor that emerges.

The following video provides a clear, step-by-step guide with several examples.

Factor By Grouping Polynomials - 4 Terms, Trinomials - 3 Terms, Algebra 2

This video from The Organic Chemistry Tutor demonstrates the factoring by grouping method very effectively.

Watch the first three examples, from the beginning until timestamp 06:31. The first example shows the basic method, the second handles negative signs, and the third involves multiple variables. Note how the process relies on finding a GCF for pairs of terms.

The key steps for factoring by grouping are:

Arrange the polynomial into two groups of two terms.
Factor out the GCF from the first group.
Factor out the GCF from the second group.
If the remaining binomials in the parentheses are identical, factor out this common binomial.

Test your understanding!

Factor the following polynomial by grouping: $2 x^{3} + 5 x^{2} - 6 x - 15$

Show answer

Let's walk through it step-by-step:

Group the terms:
$(2 x^{3} + 5 x^{2}) + (- 6 x - 15)$
Factor the GCF from each group:
- From the first group, the GCF is $x^{2}$ . Factoring it out gives: $x^{2} (2 x + 5)$ .
- From the second group, the GCF is $- 3$ . Factoring it out gives: $- 3 (2 x + 5)$ .
- Notice we factored out a $- 3$ instead of a $+ 3$ . This is crucial to make the binomial inside the parentheses match the first one.
Combine and factor out the common binomial:
$x^{2} (2 x + 5) - 3 (2 x + 5)$
The common binomial is $(2 x + 5)$ . Factoring it out gives:
$(2 x + 5) (x^{2} - 3)$

So, the fully factored form is $(2 x + 5) (x^{2} - 3)$ .

3. Special Form 1: Difference of Squares

Some polynomials have special structures that allow for quick factoring. The first of these is the difference of two squares. Any polynomial of the form $a^{2} - b^{2}$ can be factored directly. You've requested to understand why formulas work, and the next resource provides both an algebraic and a beautiful geometric explanation for this pattern.

Difference of Squares and Cubes

This video by Eddie Woo offers fantastic insight into the difference of squares formula.

Watch from the beginning until timestamp 05:36. The first part (until 02:33) shows an algebraic proof. The second part (from 02:33 to 05:36) presents a visual, geometric proof by manipulating areas, which is a powerful way to understand why the formula holds true.

The formula is:
$a^{2} - b^{2} = (a - b) (a + b)$

To use it, you just need to identify the terms that are being squared. For example, to factor $9 x^{2} - 49$ :

Recognize that $9 x^{2} = (3 x)^{2}$ and $49 = 7^{2}$ .
Here, $a = 3 x$ and $b = 7$ .
Apply the formula: $(3 x - 7) (3 x + 7)$ .

Important Note: A sum of two squares, $a^{2} + b^{2}$ , cannot be factored using real numbers. This is a common point of confusion, so remember this distinction.

4. Special Form 2: Sum and Difference of Cubes

Similar to the difference of squares, there are formulas for factoring the sum of cubes and the difference of cubes. These apply to binomials of the form $a^{3} + b^{3}$ or $a^{3} - b^{3}$ .

Difference of Squares and Cubes

Let's continue with the same Eddie Woo video, as he introduces the formulas for cubes and provides a great mnemonic to remember the signs.

Watch from 05:36 to 08:16. Pay close attention to the acronym he introduces to remember the signs in the factored form: 'Same, Opposite, Always Positive'. After watching, try his suggestion and verify one of the formulas by multiplying the factored form back out to see if you get the original expression.

Here are the formulas, which are also summarized beautifully in the image below:

Sum of Cubes: $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$
Difference of Cubes: $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$

The acronym SOAP helps with the signs in the factored form: Same, Opposite, Always Positive.

The first sign is the Same as in the original expression.
The second sign is the Opposite of the original expression.
The third sign is Always Positive.

Sum and Difference of Two Cubes Formulas with Examples — This image provides a clear summary of the formulas for factoring the sum and difference of two cubes, along with two worked examples, `x^3 + 27` and `27y^3 - 64`, illustrating how to identify `a` and `b` and apply the formulas.

Unlike the trinomial from the difference of squares, the trinomial part of a factored cube expression ( $a^{2} - ab + b^{2}$ or $a^{2} + ab + b^{2}$ ) cannot be factored further using real numbers.

Test your understanding!

Factor the following two polynomials:

$y^{3} - 64$
$8 z^{3} + 125$

Show answer

For $y^{3} - 64$ :
- This is a difference of cubes.
- Identify $a$ and $b$ : $y^{3} = (y)^{3}$ and $64 = (4)^{3}$ . So, $a = y$ and $b = 4$ .
- Apply the formula $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$ :
$y^{3} - 64 = (y - 4) (y^{2} + 4 y + 16)$
For $8 z^{3} + 125$ :
- This is a sum of cubes.
- Identify $a$ and $b$ : $8 z^{3} = (2 z)^{3}$ and $125 = (5)^{3}$ . So, $a = 2 z$ and $b = 5$ .
- Apply the formula $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$ :
$8 z^{3} + 125 = (2 z + 5) ((2 z)^{2} - (2 z) (5) + 5^{2})$ $= (2 z + 5) (4 z^{2} - 10 z + 25)$

Conclusion

Excellent work today! We've laid the first foundational stone in your journey by exploring essential polynomial factoring techniques.

Key Takeaways:

Factoring is the process of finding the simpler expressions that multiply together to form a given polynomial.
Always start by looking for the Greatest Common Factor (GCF).
For four-term polynomials, try factoring by grouping.
Recognize and apply the special formulas for difference of squares ( $a^{2} - b^{2}$ ) and the sum/difference of cubes ( $a^{3} \pm b^{3}$ ).
Use the SOAP mnemonic (Same, Opposite, Always Positive) to remember the signs for factoring cubes.

These methods are the first tools in your algebraic toolkit. In our next lesson, we will learn about polynomial long division and synthetic division. These techniques will allow us to break down more complex polynomials, especially when we suspect a certain factor but can't find it using the methods from today.

Recommended resources

1. What is Factoring? The Greatest Common Factor (GCF)

2. Factoring by Grouping

3. Special Form 1: Difference of Squares

4. Special Form 2: Sum and Difference of Cubes

Conclusion