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What is Factoring in Algebra? - Definition & Example

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  • 0:03 Parts to a Whole
  • 0:50 Factoring
  • 2:46 Practice
  • 4:57 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Factoring with ordinary numbers involves knowing that 6 is the product of 2 and 3. But what about factoring in algebra? In this lesson, we'll learn the essential elements of algebra factoring.

Parts to a Whole

Factoring in algebra is a lot like baking. I see you have some cake. Is it your birthday? Oh, I'm sorry - I didn't get you anything. But since I'm here, can I have some?

Mmm, I can really taste the baking soda. Can't you? No? Well there's definitely baking soda in this cake. OK, I can't taste it. That's because it's been combined with other ingredients to form something new - yummy cake.

In algebra, we take expressions and stir them together to make new expressions. But even though it may not be obvious what terms we started with, those ingredients are still there.

You can't take the baking soda out of a finished cake, but you can factor the original terms out of an expression. Let's learn how.

Factoring

Here is an expression: 4x - 8. Let's say we want to factor it. We can define factoring as finding the terms that are multiplied together to get an expression.

Our expression here has some important parts, like the ingredients we bake with. First, we have two terms: 4x and 8. The terms are the numbers, variables or numbers and variables that are multiplied together. Terms are separated by plus or minus signs.

8 is just a constant, or a number that is what it is. It's constantly 8. x is a variable, or a symbol standing in for a number we don't know. 4 is a coefficient. Notice the prefix 'co-.' The coefficient multiplies a variable. It's a codependent, cooperating coefficient.

Now, we need to find a factor. This is like looking for the baking soda, but usually it's a bit easier. It's more like picking the raisins out of an oatmeal cookie. You may get your hands a little dirty, but your cookie will be less raisin-y for sure.

A factor is a term that can be extracted from the equation. Think about the number 6. Its factors are 2 and 3. Why? Because 2 * 3 is 6.

With 4x - 8, we can extract a 4. Each term is a multiple of 4. If we factor out a 4, we have 4(x - 2). Note that we can reverse what we just did. 4 * x is 4x. And 4 * 2 is 8, getting us back to 4x - 8.

What if we had 3x - 8? Is there a common factor? No. 3 and 8 are what we call relatively prime. Remember that a prime number has no factors other than 1 and itself. Relatively prime numbers have no shared factors other than 1.

Practice

Let's practice factoring. Here's an expression: 6y^2 - 11y. OK, what are the factors of 6? 2 and 3. What about 11? 11 is prime. So 6 and 11 are relatively prime. Does that mean we can't factor? Are we going to have to eat the raisins? No! Look at the variables. y^2 and y. We can factor out a y. That gets us y(6y - 11). Cookie crisis averted.

To check our work, let's put it back together. y * 6y is 6y^2. y * 11 is 11y. That gets us 6y^2 - 11y. Excellent!

Let's try another: 12x^3 + 18x. OK, you might immediately see that we can factor out an x. It's like seeing unwanted olives on a pizza; they stand out, don't they? But what else? 12 and 18 share a few factors. 12's factors include 3, 4 and 6. 18's factors include 3, 6 and 9. Where's the overlap? 3 and 6.

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