Factor Quadratics when a = 1

In this module, we are going to learn how to factor which is defined as un-distributing. Recall that you know that to distribute means to multiply a value to each part:

3(x + 4) = 3(x) + 3(4) = 3x + 12

But when we factor, we must un-distribute. Here are some examples:

Expanded	Factored
3x - 18	3(x - 6)
4xy + 5y	y(4x + 5)
x² + 5x	x(x + 5)
-2x² + 4x + 2	-2(x² - 2x - 1)
3(x - 1) + 5y(x - 1)	(x-1)(3 + 5y)

We often refer to the process above as factoring out the Greatest Common Factor (GCF).

Note: If the leading coefficient is negative, factor out a negative GCF!

Factoring the GCF Practice

Factor out the GCF of the expressions below.

3xy + 9y
5x² +15x -25
x³ - 2x² + 3x
-5x² +5
2x(x + 5) + 3(x + 5)

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE.

Now let's look at quadratic equations. Recall earlier in this unit we learned how to multiply polynomials. We distribute the terms in each binomial to the other terms:

(x + 5)(x - 3)

x² - 3x + 5x - 15

x² + 2x - 15

First, we are going to start with quadratics that have a leading coefficient of 1, or a = 1. Let's multiply another set of binomials and see if we notice any patterns:

(x - 4)(x + 3)

x² + 3x - 4x - 12

x² - x - 12

Notice that in both examples, the constants in the binomials add to give us the "b" term and multiply to give us the "c" term.

How to factor explanation

Let's try one:

x² - 10 x + 21

(x_____7)(x_____3)

(x - 7)(x - 3)

Think of factors of 21 that would add to 10. Now determine which signs would make -10 and 21. You have factored!

Now you can check your work by "expanding" the quadratic:

(x - 7)(x - 3) = x² - 7x - 3x + 21 = x² - 10x + 21

So we know we got it right!

Sometimes quadratic expressions are not factorable, when that happens we call those expressions prime!

At times, you'll come across quadratic equations that have a leading coefficient that is not 1. When you see this, you want to see if you can factor that value out of the entire expression.

2x² - 8x + 6

2(x² - 4x + 3)

2(x - 3)(x - 1)

Factor out the GCF of 2 from each term. Now try and factor what is left. Final answer!

Factoring Practice

x² + 5x + 6
x² - 4x - 12
x² - x - 30
x² + 11x - 26
x² + 4x + 1
x² - 49
3x² + 21x + 36
4x² - 8x - 12
-2x² - 24x - 22
-x² + 9x - 14

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE.

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