Lesson 79: Factoring Trinomials by Using the GCF

New Concept

The terms of a polynomial that is factored completely will have no common factors other than 1.

What’s next

Next, you will apply this principle to factor various trinomials, starting by identifying and extracting the greatest common factor (GCF).

Property

To factor a polynomial completely, first identify and factor out the Greatest Common Factor (GCF) from all terms. The GCF is the largest monomial that divides into each term of the polynomial.

Explanation

Think of the GCF as the biggest shared ingredient in every term. Before you start solving the main trinomial puzzle, pull that GCF out to the front! This step simplifies the remaining expression, making it much smaller and easier to work with. It's like tidying up your desk before starting homework—it just makes everything clearer and simpler.

Examples

$x^5 + 6x^4 + 8x^3 = x^3(x^2 + 6x + 8) = x^3(x + 2)(x + 4)$
$5x^3 - 10x^2 - 120x = 5x(x^2 - 2x - 24) = 5x(x + 4)(x - 6)$

Always look for a common factor first; it simplifies the entire puzzle. This example shows how to use the greatest common factor (GCF) to make factoring easier.

Example Problem

Factor the expression $3x^3 + 9x^2 - 30x$ completely.

Step-by-Step

1. First, find the greatest common factor (GCF) of all the terms in the polynomial. The GCF of $3x^3$, $9x^2$, and $-30x$ is $3x$.
2. Factor out the GCF from the expression:

3. Now, focus on the trinomial inside the parentheses, $x^2 + 3x - 10$. Find two numbers that have a product of $-10$ and a sum of $3$.

4. Use these two numbers, $5$ and $-2$, to write the trinomial as a product of two binomials.

Property

When the leading coefficient of a trinomial is negative, it is standard practice to factor out a $-1$ along with any other common factors.

Explanation

Nobody likes starting a problem with a negative attitude! If your polynomial's first term is grumpy and negative, factor out a $-1$ to flip all the signs inside the parentheses. This makes the new leading term positive and cheerful, which makes finding your factor pairs so much simpler. It's a quick trick to turn a frown upside down!

Examples

$-x^2 + 2x + 48 = -1(x^2 - 2x - 48) = -(x + 6)(x - 8)$
$-2x^3 - 10x^2 + 48x = -2x(x^2 + 5x - 24) = -2x(x + 8)(x - 3)$

When the leading term is negative, it's a sign to pull out the negative along with the GCF. Let's see how this works.

Example Problem

Factor the expression $-2x^3 - 10x^2 + 48x$ completely.

Step-by-Step

1. Notice the leading coefficient is negative ($-2$). The GCF of the terms is $2x$. To make the new leading term positive, we factor out $-2x$.
2. Factor out the negative GCF from the expression. Remember that dividing by a negative changes the signs of the remaining terms.

3. Now, factor the trinomial $x^2 + 5x - 24$. We need two numbers that have a product of $-24$ and a sum of $5$.

4. Use these numbers, $8$ and $-3$, to write the final factored form, keeping the GCF out front.

Property

For polynomials with multiple variables, the GCF may include variables as well as numbers. Factor out the entire GCF first to simplify the expression.

Explanation

Don't be scared when you see more letters in the mix! The GCF can include variables like $a$, $b$, or $k$ if they appear in every single term. Just find all the common ingredients, both numbers and variables, and pull them out together. The trinomial that's left over will behave just like the simpler ones you know.

Examples

$ax^4 + 10ax^3 + 21ax^2 = ax^2(x^2 + 10x + 21) = ax^2(x + 3)(x + 7)$
$7ky^3 + 14ky^2 - 105ky = 7ky(y^2 + 2y - 15) = 7ky(y + 5)(y - 3)$