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A Quadratic Equation

Let's see if we can use our knowledge of the greatest common factor to solve this new equation. What's the greatest common factor of x squared, 5x, and 6? Now, this question is pretty tough, so take some time to think about it, and think about any number that can go in to each of the terms. Writing out the prime factorizations of each of these might help as well.

A Quadratic Equation

The greatest common factor, or the GCF is 1. These would be the prime factors for X squared, 5 X and 6. Notice how I have it listed all three of them don't share a common factor, but we really know that we can multiply each of these by think that a group of terms doesn't have a common factor they actually do, and it would be 1.

Greatest Common Factor

Whenever we start to factor an expression like this, we should always take out the greatest common factor first. Factoring out a one in this case, will not change our equation. So, we need to learn something else in order to solve this problem.

Factor the GCF

We are going to learn a factoring method called factoring by grouping. We're going to start my grouping terms together that have common factors. We are going to trying to remove our common factors from this group of terms and from this group of terms of so what's the greatest common factor for ax and ay and what's the greatest common factor for 2x and 2y, you can write your answers here We're going to start my grouping terms together that have common factors.

Factor the GCF

Looking at these two terms, we can see that they both share a factor of a. So a is the greatest common factor. For the next two terms, they only share a common factor of two.

Another Common Factor

We can factor an a from ax and ay so we have a times x plus y. We can also factor a 2 from 2x and 2y. So we'll have plus 2 times x plus y. And notice if we were to just distribute we'd get back with what we started with. But in this line there's something incredible happening. This would be one term made up of two factors a times x plus y. This would be another term made up of two factors, terms? Write your answer here.

Another Common Factor

Well, they both share a factor of x plus y. Remember that a here is one factor, and the quantity x plus y is another factor, since we're multiplying these two numbers together. The same is true on the right, 2 is a factor, and then x plus y is another factor.

Factoring by Grouping

Since this first term and the second term share an x plus y, we can actually factor it out. This is called factoring by grouping.

Factor by Grouping Check

We can perform this because we were dealing with factors here. We were multiplying between these two numbers. So we removed a common factor of x plus y. And notice we still have multiplication here. Again if I think about distributing this a to x plus y, I would get this first term, and if I distributed this positive 2 to this x plus y, I would get my second term. This factoring is the opposite, or the inverse, of the distributive property. So you try one out. What would be the factored form for this expression? Write your answer here. Be sure you use parentheses, and there's no need to put a multiplication sign in between them.

Factor by Grouping Check

Here's the correct answer, and great job if you got it right. We can factor a 2 a from the first two terms to give us 2 a times a plus 4. The second two terms share a b, so we can factor that out, leaving us with b times a plus 4. Now this term and this term share an a plus 4, so we can factor that out, leaving us with

Negative GCF

Try factoring these four terms. Notice that the first two terms are positive, but the second two terms are negative. This means that for these two terms, you should have a negative greatest common factor. Try using our same approach, and just be careful with the negative sign. Factor this expression and write your answer here.

Negative GCF

The quantity 2y minus 3z times the quantity 3x plus 4 is correct. Fantastic work for getting that one right. Now, I know we haven't covered negative signs yet, so don't worry if you didn't get it right. Let's see how we could do this. I can see that both this term and this term share a y, and also, they're both divisible by 2. So, 2y must by my greatest common factor. If I divide 2y into Here, this term is negative and this term is negative. The negative 9 and negative 12 share a negative 3 as a common factor. They also share a z. Now, I want to think about what two terms should go in here. What number times negative number times negative 3z will give me negative 12z. This must be positive 4. And this is great. This first term and the second term share a common factor of 3x plus 4, so we can factor again. So, this 3x plus 4 is here for one of my factors and 2y minus 3z is my other factor. And again, this line should make sense because if we distribute 2y, we'll get 2y times 3x plus 4, our first term. And then if we distribute negative 3z, we'll have negative 3z times 3x plus 4, our second term. So, this is our last finaled factored form.

Rearrange Terms

But sometimes factoring by grouping isn't so straightforward. Let's consider an expression like this. Notice that in these first two group of terms I have a greatest common factor of one. The same is true in these second two terms. So I need to do something else. So we can switch the order of the terms. So you try this one. Regroup the terms and then try factoring by grouping.

Rearrange Terms

These would be my two factors, excellent work if you got this one correct. You might have done this in a couple of ways, let's check out one of the methods. If we switch the middle two terms we can see that we'll get a common factor of N in the first terms and a common factor by P in the second terms. If we factor a 7 N from the first two terms we'll be left with two M and positive one. And if we factor a 5p from the second term, we'll be left with 1 and positive 2m. These 2 factors might appear different but we just wanted to switch the order of these 2 terms. Remember addition is commutative. Now that we can see there's a common factor of 2m plus 1 in the first term and 2m plus 1 in the second term. We factor again which leaves us with our factored form, 7n plus 5p times 2m plus 1. But we could have regrouped the terms in another way. I could have grouped this first term and this last term together and then kept the middle 2 terms together as well. If we factor a 2m from these first 2 terms we'll be left with 7n and They don't share any variable factors or number factors. Notice that I have 7n plus 5p and 7n plus 5p in these two parenthesis. This is the next common factor. So, our two factors are 2m plus 1 and 7n plus 5p. Notice that these answers are exactly the same. We've just switched the order of the multiplication. This first parenthesis represents a number. And the second parenthesis also represents a number. So, it would be like 2 times 3, would be the same as 3 times 2. Our multiplication is commutative.

Number Puzzles Again

Before we get to the practice for this lesson, let's try some more number puzzles. You'll discover the importance of these number puzzles, in the upcoming lessons.

Number Puzzles Again

Here are the factored pairs for negative 45 and we want to find one that sums to four. Well, negative 5 and 9 multiply to negative 45 and add to 4, so these are my answers for the first number puzzle. For the second number puzzle, I want to find 2 numbers that multiply to negative 45 and add to negative 12. That must be that sum to negative 14. Well, here are the positive factor pairs for positive negative factor pairs of positive 45. We know a negative times a negative is a positive. So now, I'm just looking for the factor pair that adds to negative 14. This would be negative 5 and negative 9. This must mean the last factor pair is negative 1 and negative 45. They sum to negative 46 and they multiply to positive 45.

Number Puzzle

Before we get to the practice for this lesson, lets try some more number puzzles. You'll discover the importance of these number puzzles in the upcoming lessons. Here are the factor pairs for negative 45, and we want to find one that sums to 4. Well, negative 5 and 9 multipliy to negative 45 and add to 4. So these are my answers for the first number puzzle. For the second number puzzle, I want to find two numbers that multiply to negative 45 and add to negative 12. That must be 3 and negative 15. For the third number puzzle, I want to find factors of 45 that sum to negative 14. Well, here are the positive factor pairs for positive 45, but none of these pairs add to negative 14. This must mean I need the negative factor pairs of positive 45. We know a negative times a negative is a positive, so now i'm just looking for the factor pair that adds to negative 14. This would be negative 5 and negative 9. This must mean the last factor pair is negative 1 and negative 45. They sum to negative 46 and they multiply to positve 45

Patterns of Factors

Let's look more closely at these two number puzzles. I put the first two here, and the second two here. In our number puzzles, these were the products. So explain the observation: if the product is negative, what's true about the factors, and if the product is positive, what's true about these factors? Check all the statements that are true in each case.

Patterns of Factors

Well, if the product is negative then one of the factors must be negative, that would be this case. If the product is positive then that means that both the factors can be negative, that's this case, but we want to be careful over here. We could have made another number puzzle where the product was positive 45 and the sum was positive 14. In this case both of the factors would have been positive. This case could also be true. So we have to have one negative factor if our product is negative. And we can either have two positive factors or two negative factors if our product is positive.

Factor by Grouping Practice 1

Try using your knowledge of the greatest common factor, and factoring by grouping, to factor this expression. Good luck and have fun.

Factor by Grouping Practice 1

We can factor a 2x from the first two terms and a positive 5 from the second two terms. We can see that this first term and the second term have a common factor of 3x plus 2. We factor that out leaving us with 2x plus 5 times the quantity 3x plus 2. So here's our factored expression.

Factor by Grouping Practice 2

Try factoring these four terms. This one's a bit harder since we have negative signs.

Factor by Grouping Practice 2

Great factoring skills if you got this correct. We can factor a 3x from the first two terms. We'll have 3x times the quantity 4x minus 3. I've been saying this word quantity a lot and I use it when I really talk about the parentheses. I have 3x times some quantity 4x minus 3. If we distribute the 3x to 4x and minus 3, we wind up with what we started with. So this is correct factoring. Next, I can see that negative 8x and 6 share a factor of 2. This would leave me with negative 4x plus 3 in the second parentheses. You might have gotten stuck here and that's okay. We notice that this first parentheses and the second parentheses aren't the same. The 4x is positive and this is negative. The 3 is negative, but this 3 is positive. Well, instead of taking out a positive 2, let's take out a negative 2. If we factor out a negative 2 from negative 8x, we're left with positive 4x. And if we factor a negative 2 from positive 6, we're left with negative 3. Finally, we factor out the 4x minus 3 from the first term and the second term. This leaves us with 3x minus 2 times 4x minus 3. You could have either have written your answer like this or like this. Both are correct.

Factor by Grouping Practice 3

Try factoring these four terms. Watch out for negative signs and remember to rearrange terms if necessary.

Factor by Grouping Practice 3

There are a number of ways to do this problem. I'm going to show you one of them. I can factor a 6a from the first two terms, leaving me with 4a minus bc in the parentheses. For the second parentheses, I could try pulling out a positive positive 4a here. So instead of factoring a positive 5, let's factor a negative you want to be really careful and check. If we distribute both of the terms, we should wind up with we started with. Most importantly, I'm concerned about this last term. I have negative 5 times negative bc. A negative times a negative is a positive. So yes, this is correct. Finally, we factor out our 4a minus bc, which leaves us with this expression. So these are my two factors, and this is the expression in factored form.

Factor by Grouping Practice 4

Now that we've learned about factoring by grouping, let's try another practice problem. Here, we have four terms one, two, three, four, 18x squared minus 9xy minus 30x plus 15y. Since there is four terms, we know that we're going to be factoring by grouping and I want you to put your answer in this box. Don't forget to factor out the GCF if you can find one.

Factor by Grouping Practice 4

So, how did you do? The correct answer is 3 times 2x minus y times 3x minus 5. Let's see how we got that answer. Whenever I see a four term polynomial, I know I want to use factoring by grouping. But remember the first step in any factoring problem is to look for a GCF. In this case, when I look at the four terms, I know that it has a GCF of 3. So I'm going to factor out the GCF from these four terms. So, when I factor out a 3, I'm left with 6x squared minus 3xy minus 10x plus 5y. Now, I'm going to try to factor out what's left inside the parentheses. When I factor by grouping, I want to find the GCF of the first 2 terms, which is 6x squared minus 3xy. The GCF of these first 2 terms is 3x. When I factor a 3x out, I'm left with 2x. 3x times 2x is 6x squared minus y. 3x times negative y gives me negative 3xy. The next thing I want to do is factor out the GCF of the second 2 terms. Now, you have to be careful here. We know we want the second binomial to be 2x minus y over here. But right now, we have negative 10x plus y. This is one of the cases we need to factor out the negative. The GCF of because I want positive 2x. Negative 5 times 2x gives me negative 10x and negative 5 times negative y gives me 5y. Because negative 5 times 2x is negative factor out the GCF of the two terms inside the parentheses. Remember to carry along the three, and I'm left with 2x minus y, times 3x minus 5. And these second pair of parentheses is redundant. So I can just write 3 times 2x minus y times 3x minus 5. Did you get that? Well done. Let's learn more about factoring.