Factoring Polynomials
2026-02-28 15:52 Diff
https://brightchamps.com/en-us/math/algebra/factoring-polynomials

In this section, we will focus on three main methods used to solve polynomials.

Greatest Common Factor

In this method, we take out the largest number (and variable, if any) that all the terms have in common. It’s like reversing the distributive property.

Distributive property: 

p(q + r) = pq + pr

Factored form:

pq + pr = p(q + r), where p is the greatest common factor.

Example: Factor the polynomial 12x + 18

Step 1: Find the GCF 

Break each term into its factors:

\(12x = 3 × 4 × x\) or \(6 × 2 × x\)

\(18 = 3 × 6\)

So, the GCF is 6.

Step 2: Factor it out

\(12x + 18 = 6(2x + 3)\)

Step 3: Check by distributing

\(6(2x + 3) = 6 \times 2x + 6 \times 3 = 12x + 18\)

Factoring polynomials by grouping 

This method works well while trying to factor trinomials (three-term expressions) that can’t be factored just by taking out a common factor. The idea is to split the middle term into two parts so we can group the terms and factor in pairs.

Example: Factor \(x^2 + 11x + 24 = (x + 3)(x + 8)\)

Step 1: Find two numbers

Look for two numbers that:

Add up to 11 (the middle term), and

Multiply by 24 (the last term)

The numbers 3 and 8 work because:

\(3 + 8 = 11\)

\(3 × 8 = 24\)

Step 2: Rewrite the middle term using these two numbers

\(x^2 + 11x + 24 = x^2 + 3x + 8x + 24\)

Step 3: group the terms and factor

\((x^2 + 3x) + (8x + 24)\)

\(x(x + 3) + 8(x + 3) \)

Take the common binomial factor, 

\((x + 3)(x + 8)\)

So, \((x + 3)(x + 8)\) are the factors of \(x^2 + 11x + 24\)

Factoring using Identities

Algebraic identities can also help in factorizing polynomials. Commonly used identities for this purpose are:

\((a + b)^2 = a^2 + 2ab + b^2\)

\((a - b)^2 = a^2 - 2ab + b^2\)

\(a^2 - b^2 = (a + b)(a - b)\)

Let’s take examples for each identity listed above.

Example 1: Factorize \(x^2 + 6x + 9\), using (a + b)2

Step 1 - Compare with the identity

\(x^2 + 6x + 9 = x^2 + 2 \times 3 \times x + 3^2\)

We can see the pattern a2 + 2ab + b2, where a = x and b = 3

Step 2 - Apply the identity

\(x^2 + 6x + 9 = (x + 3)^2\)

So, (x + 3)2 is the answer.

Example 2: Factor \(x^2 - 10x + 25\) using identity (a - b)2

Step 1 - Match with the identity

\(x^2 - 10x + 25 = x^2 - 2 \times 5 \times x + 5^2\)

Here, a = x and b = 5

Step 2 - Apply the identity

\(x^2 - 10x + 25 = (x - 5)^2\)

(x - 5)2 is the answer.

Example 3: Factor x2 - 49 using identity a2 - b2

Step 1 - It needs to be the difference between two perfect squares

\(x^2 - 49 = x^2 - 7^2\)

Step 2 - Apply Identity a2 - b2 = (a + b)(a - b)

So, \(x^2 - 49 = (x + 7)(x - 7)\)