Factoring Polynomials

Easiest, Step by Step Method for Factoring Polynomials 🧮

What are Polynomials? 🎈

A polynomial is a mathematical expression containing multiple terms. For example, the following expressions are polynomials:

The individual terms are usually written in order of decreasing degree. The degree is the largest exponent of a term or expression. For example, the degree of the first polynomial (x^4 + 3x^3 + x^2 + 4x + 5) is 4, because that’s the largest exponent of any term in the expression. The degree of a single term like x^2 would be 2. The degree of x^2 + 5 would also be 2. You may see this also referred to as the “order” of an expression/term; the terms “degree” and “order” are interchangeable in Algebra. 😎

💡 “Factoring polynomials” means…

Factoring polynomials is a way to rewrite them in a simpler format, breaking them down into smaller components. We are answering this question: what do you have to multiply to get this expression? For example, x2 + 3x + 1 becomes (x+2)(x+1). 👍

Common Methods 🔑

1️⃣ Factor by isolating the greatest common factor (GCF)

Sometimes, this is the only thing you can do to factor. You could go further and try to get more factors out of an expression, but generally, you don’t need to; if you get to a point where decimal quantities are a part of your expression, like (x+3.4)(x-3.2),  you should deem that your stopping point. Rarely will problems ever require a non-whole-number answer. 💘

Example: Factor the expression 8x^4 + 10x^3 + 6x. 

What can we divide out of each coefficient (the number next to a variable, i.e. x) of each term in the expression? 2. What can we divide out of every term? x! Therefore, the greatest common factor of each term in the expression is 2x, and the factored solution is as follows: 2x (4x^2 + 5x + 3).

2️⃣ Factor by grouping

This method relies on a common factor, typically a binomial (two-term polynomial), within a polynomial once it is separated. Because not every polynomial has a common binomial once separated, this doesn’t always work, but it is worth trying if you can spot a common binomial! 🔭

Example: Factor the expression 4x^2 + 8x + 2x + 4. 

First, let’s separate the polynomial into two expressions: (4x^2 + 2x) + (8x + 4). 

We can use the same strategy of factoring out a GCF to simplify each part. (4x^2 + 2x) becomes x (4x+2) and (8x + 4) becomes 2 (4x+2). We multiplied 2 different quantities by the same factor, (4x+2): x and 2. Therefore, the expression simplified by grouping becomes (x+2)(4x+2).

3️⃣ Guess-and-check

This method works mainly for simpler 3-term AKA quadratic polynomials. Still, if you have times tables of larger numbers memorized or have practiced factoring polynomials more, you can use it on more complicated quadratics or even cubic polynomials. 

Example 1: Factor x^2 - 3x - 2

While guess-and-checking, we must look for a term that adds to the middle term's coefficient and has a product of the last term of a trinomial/quadratic (three-term) expression. In this case, what numbers add to -3 and yield a product of -2? -2 and -1 do.

Those are the numbers we insert into the format of factors of a quadratic equation (x-(factor))(x-(factor)): (x-2)(x-1).

Example 2: Factor x^2 - x - 12. 

What numbers add to -1 and yield a product of -12? -4 and 3, so we insert them into the aforementioned mold to get an answer of (x+3)(x-4)!

Special Cases: Formulas to Know 👨‍💼

Special cases of polynomials have formulas that always work. Using them can save time:

🟩 Difference of Squares 

One of the most common special cases is the difference of squares, the third formula above: a^2-b^2 = (a+b)(a-b). x^2-4 qualifies as a difference of squares. 

Of course, the b-term in the difference of squares is not explicitly written; you can see that 4 = 2^2 and that 2 is, therefore, the b-term in this expression. Thus, the solution is (x+2)(x-2). 

🧊 Difference and Sum of Cubes 

The formula for a difference of cubic expressions or “cubes” is a^3-b^3 = (a-b)(a^2+ab+b^2). For a sum of cubes, the formula isn’t much different: (a+b)(a^2-ab+b^2). One way to differentiate between the two formulas is to recall that the sum of cubes formula has alternating signs: +, -, +, whereas the difference of cubes is -, +, +. 

❗ Other Special Cases 

The first special case listed in the image above is a^2+2ab+b^2 = (a+b)^2. For example, 9x^2+24x+16 rewritten as (3x)^2 + (23x4) + 4^2 leads us to (3x+4)^2. 

The second special case listed is a^2-2ab-b^2 = (a-b)^2. The expression 16x^2+16x+4 fits this special case. Rewritten to fit the formula, it becomes (4x)^2 + (24x2) + (2^2). Our final answer is then (4x+2)^2!