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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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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.

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.

Here is an expression: 4*x* - 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: 4*x* 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 4*x* - 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 4*x*. And 4 * 2 is 8, getting us back to 4*x* - 8.

What if we had 3*x* - 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.

Let's practice factoring. Here's an expression: 6*y*^2 - 11*y*. 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*(6*y* - 11). Cookie crisis averted.

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

Let's try another: 12*x*^3 + 18*x*. 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.

What should we factor out? If we factor out the 3, we'd have 4 and 6. That's better, but that pizza still has olives, so to speak. We want the greatest common factor. That's simply the biggest shared factor. Here, that's 6. If we factor out 6*x*, we get 6*x*(2*x*^2 + 3).

Let's check. 6*x* * 2*x*^2 is 12*x*^3. That's good. And 6*x* * 3 is 18*x*. Good again. This pizza is safe to eat!

How about one more? 5*y* + 5. Well, 5 and 5, what do we do here? We can extract that 5. Then we get 5 times *y* plus what? What times 5 is 5? 1. So it's 5(*y* + 1). That's a bit like asking for an ice cream cone, minus the ice cream.

To summarize, we learned about factoring in algebra. To factor is to find the terms that are multiplied together to make an expression.

Expressions consist of various terms. A term can have certain parts, like constants, variables and coefficients.

When we factor in algebra, we're looking for the greatest common factors that are shared by the terms in an expression. If we only have numbers that are relatively prime, like 7 and 9, then we can't factor out any constants. But if there are common factors, then we pluck them like olives from a pizza. Just remember to check for ones hiding beneath the cheese!

Once you've finished checking out this lesson, you might be prepared to:

- Explain the process of factoring
- Identify common factors
- Factor an algebraic expression

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- What is Factoring in Algebra? - Definition & Example 5:32
- Using Prime Factorizations to Find the Least Common Multiples 7:28
- Equivalent Expressions and Fraction Notation 5:46
- Using Fraction Notation: Addition, Subtraction, Multiplication & Division 6:12
- Factoring Out Variables: Instructions & Examples 6:46
- Combining Numbers and Variables When Factoring 6:35
- Transforming Factoring Into A Division Problem 5:11
- Factoring By Grouping: Steps, Verification & Examples 7:46
- Go to High School Algebra: Factoring

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