In this video, we're going to talk about factoring. How to factor the greatest common factor, the GCF. So let's say if we have the expression 3x plus 15. What is the greatest common factor between 3x and 15? 3x and 15 are both divisible by what number? They're both divisible by 3, so therefore 3 is the GCF, the greatest common factor.
3x divided by 3 is simply x. And 15 divided by 3 is 5. And so that's how you can factor an expression by removing the GCF, the greatest common factor. So that's the answer for the first one.
Now let's try some other examples. Try this one. 7x minus 28. Go ahead and factor this binomial by removing the GCF.
So 7x and negative 28 are both divisible by 7. 7x divided by 7 is x. Negative 28 divided by 7 is negative 4. And so that's the answer. now go ahead and try these two problems 4x squared plus 8x and also 5x squared minus 15x cube so 4x squared plus 8x they're divisible by what they're both divisible by what term Both are divisible by 4, and they're both divisible by x.
So 4x is the GCF. 4x squared divided by 4x is simply x. 8x divided by 4x is 2. And so that's the solution for the first one. Now for the next one, 5x squared and 15x cubed are both divisible by 5x squared.
What you do is, for the x variables, you choose the smaller of the two. So let's take out 5x squared. 5x squared divided by 5x squared is 1. Negative 15x cubed divided by 5x squared is negative 3x.
5 times negative 3 is negative 15. x squared times x is x cubed. You can always multiply to check your work. If you distribute... This is going to be, you should get the original expression. So that's how you can tell if your work is correct.
In this lesson, we're going to talk about factoring by grouping. Typically, you'll have a polynomial expression. with four terms, like this one, x cubed minus 4x squared plus 3x minus 12. Now, in order to factor by grouping, you want to separate it into two parts. The first two terms take out the GCF, the greatest common factor. The greatest common factor between x cubed and 4x squared is x squared.
x cubed divided by x squared is x. Negative 4x squared divided by x squared is negative 4. Now, do the same for the last two. Take out the GCF.
The GCF between 3x and negative 12 is stream. 3x divided by 3 is x. Negative 12 divided by 3 is negative 4. Now, if you get this common term, that means you're on the right track. At this point, you can write two parentheses. What's going to go in the first one is the x minus 4. What's going to go in the second one is what you see on the outside, x squared plus 3. And so, that's going to be the answer.
Now let's try another example. 2r cubed minus 6r squared plus 5r minus 15. Feel free to pause the video and factor by grouping. So let's start.
with the first two terms the GCF is 2r squared 2r cubed divided by 2r squared is r negative 6r squared divided by 2r squared is negative 3 now for the last two terms the GCF So if we take out a 5, 5r divided by 5 is r, negative 15 divided by 5 is negative 3. By the way, how can you know if you can factor by grouping? Because sometimes you can, and sometimes you can't. If you divide negative 6r squared, or rather, just look at the coefficients. If you divide negative 6 by positive 2, notice that you'll get negative 3. And if you divide negative 15 by 5... you will also get negative 3. When you see that, when the first two terms have the same ratio as the last two terms, that means that you can factor by grouping.
Now, in the next step, let's factor out r-3. If we take out r-3 from the first term, we're going to have 2r squared left over. And if we take this term, divided by the GCF, r minus 3, it's going to give us plus 5. And so, this is the solution. So now you know how to factor by grouping. In this video, we're going to talk about factoring trinomials.
Particularly when the leading coefficient is 1. So here's an example. x squared plus 7x plus 12. The leading coefficient is 1. Now for these examples, all you need to do is find two numbers that multiply to 12, but that add to the middle term 7. So let's make a list. Factors of 12 are 1 and 12. 12 divided by 2 is 6. 12 divided by 3 is 4. 1 plus 12 is 13, 2 plus 6 is 8, but 3 plus 4 is 7. So to factor it, all it is is just going to be x plus 3 times x plus 4. If you FOIL this expression, you will get the original expression.
X times x is x squared, x times 4 is 4x, 3 times x is 3x, 3 times 4 is 12, and 4x plus 3x is 7x. So, FOILing is just a reverse of factor. Now, let's try another example.
What about x squared plus 3x minus 28? Feel free to pause the video and see if you can do this one. So, what two numbers multiply to negative 28 but add to 3? Well, we know that 7 times 4 is 28. So it could be negative 7 and positive 4, or positive 7 and negative 4. Negative 7 plus 4 is negative 3. Positive 7 plus negative 4 is positive 3. So it's going to be x plus 7 times x minus 4. And that's how you can factor it.
Now let's try this one. x squared minus 3x minus 10. So what two numbers multiply to negative 10 but add to negative 3? So this is going to be negative 5 or 2, or 5 and negative 2. Now negative 5 plus 2 adds up to negative 3 and it still multiplies to negative 10. So the answer is x minus 5 times x plus 2. What about this example?
x squared minus 9x plus 20. What two numbers multiply to 20 but add to negative 9? So they both have to be negative numbers. 20 divided by negative 1 is negative 20. If we divide it by negative 2, we're going to get negative 10. 3 doesn't go into 20, but if we divide it by negative 4, we'll get negative 5. And so it's going to make a list.
Negative 4 plus negative 5 adds up to negative 9. So the answer is x minus 4 times x minus 5. Now let's try one more example. 2x squared plus 20x plus 48. Now you can use the same technique, but notice that you can take out the GCF. And if you see that, you...
We want to do that first. So let's factor out a 2. 2x squared divided by 2 is x squared. 20x divided by 2 is 10. 48 divided by 2 is 24. Now what two numbers multiply to 24 but add? So if we make a list, we have 1 and 24. If we divide it by 2, we'll get 12. If we divide it by 3, it will give us 8. If we divide it by 4, we'll get 6. 4 plus 6 is 10. So it's going to be x plus 4 times x plus 6. And with practice, you'll be able to do this quickly. Now what if the leading coefficient is not equal to 1?
What should you do? Well, what you want to do is, you need to multiply the leading coefficient by the constant term. That is, you want to multiply 2 times negative 3. 2 times negative 3 is negative 6. Next, find two numbers that multiply to negative 6, but add to negative 5. Well, we have 2 and negative 3, and we have 1 and negative 6. But 1 plus negative 6 is negative 5. So what you want to do at this point is replace negative 5x with negative 6x plus 1x, because they add up to negative 5. And now that you have four terms, factor by grouping. In the first two, take out the GCF, which is 2x.
2x squared divided by 2x is x. Negative 6x divided by 2x is negative 3. Now, in the last two terms, if there's nothing to take out, take out a 1. So, if we take out 1, it's just going to be the same, x minus 3. Now, if we factor x minus 3, which we have in both terms, we're going to be left over with 2x plus 1. And so that's how you can factor it. And if you want to, you can FOIL it to check to see if that's the right answer.
Let's try another one. 15x squared plus x minus 6. So let's multiply 15 and negative 6. 15 times negative 6 is negative 90. Now what two numbers that multiply to negative 90 add up to positive 1? So we have 1. 90 is divisible by 2 that will give you 45 if you divide it by 3 you get negative 30 for this going to 90 if you divide it by 5 you'll get negative 18 if you divide it by 6 negative 15 7 and 8 doesn't go into it. If you divide it by 9, you'll get negative 10. Notice that 9 and negative 10, they differ by 1. They add up to negative 1, so reverse the sign, and they'll add up to positive 1. So let's replace 1x with 10x and negative 9x. The order in which you place it doesn't matter.
I just prefer to keep the 15 and the 10 because it's easy to take out a 5. So now let's factor by a group. In the first two terms, let's say that I have a group of 5. So I'm going to put let's take out 5x. 15x squared divided by 5x is 3x. 10x divided by 5x is 2. Now, in the last two terms, let's take out negative 3. Negative 9x divided by negative 3 is 3x. Negative 6 divided by negative 3 is plus 2. Notice that we have a common term so now let's take out 3x plus 2 We'll be left over with 5x minus 3 and so that is the answer So now you know how to factor a trinomial when the lead coefficient is not one.
In this lesson we're going to talk about how to factor perfect square trinomials. A perfect square trinomial is in the form a squared plus 2ab plus b squared and It's equal to a plus b times a plus b, which you can write it as a plus b squared. So how can you tell if a perfect square trinomial is a perfect square trinomial? So what you want to do is take the square root of this term and the square root of that term, multiply it by 2, you should get the middle term.
So for example, x squared plus 8x plus 16. Now for this particular trinomial, We can factor it using the lessons that we've learned in 8.3. Sometimes, however, this expression may not be easy to factor, and it might be useful to use the equation. So let's do it both ways.
In lesson 8.3, we would find two numbers that multiply to 16, but that add to 8. This would be 4 and 4. 4 times 4 is 16, 4 plus 4 is 8. So it's going to be x plus 4 times x plus 4. And because they're the same, it's x plus 4 squared. So using that method... It's not bad.
You can quickly get the answer for this type of example. But now, let's use the equation. So first, take the square root of 1, the coefficient. The square root of 1 is 1. and the square root of 16 is 4. Multiply 1 and 4. 1 times 4 is 4, and then multiply it by 2. If this answer is equal to the middle coefficient, then you can factor.
It's a perfect square trinomial, so you can use the equation. That's what I meant to say. That was at a loss of words.
So the equation is a plus b squared. So all you need to do is take the square root of x squared, which is x, the square root of 16, which is 4, and just write it like this. Now let's try another example. 9x squared plus 6x plus 4. Is this a perfect square trinomial? So the square root of 9 is 3. The square root of 4 is 2. 3 times 2 is 6, and you have to double it.
6 times 2 is 12, which is not 6. So this is not a perfect square trinomial. Now what about this one? 4x squared plus 12x plus 9. The square root of 4 is 2. The square root of 9 is 3. 2 times 3 is 6. And if we double it, we do get 12. So that's a perfect square trinomial.
So to factor it... the square root of 4x squared is 2x, the square root of 9 is 3, and then it's going to be 2x plus 3 squared. Now, if we were to use the other technique, here's what we'll have to do it's going to be a lot more work in this particular example we're going to have to multiply 494 times 936 now when you find two numbers that multiply to 36 but add to 12 so we have two and 18 we have 49 there's also a 3 times 12 And as you can see, it's probably going to be a fraction. So this is a lot much, this is harder to factor.
Or, actually, 6 and 6, there we go. 6 times 6 is 36, but 6 plus 6 adds up to 12. So we can replace the middle term with 6x plus 6x. Now, let's take out the GCF in the first two terms. That's going to be 2x.
4x squared divided by 2x is 2x. 6x divided by 2x is 3. For the last two, we can take out a 3. 6x divided by 3 is 2x. 9 divided by 3 is 3. So now we can take out 2x plus 3. And we're going to be left with 2x plus 3. So as you can see for this example, it's easier to use the equation.
This is a lot more work to factor this way. Let's try another example. One more at least. Try this one. 9x squared plus 30x plus 25. So first, identify if it's a perfect square trinomial, and then factor it.
So let's take the square root of 9. Just 9. That's going to be 3, and the square root of 25 is 5. Now, 3 times 5 is 15, and 15 times 2 is 30. So, it's a perfect square trinomial. So now we can use the formula to factor it. Let's take the square root of 9x squared and 25. The square root of 9x squared is 3x. The square root of 25 is 5. And after that, just square it.
So that's how you can factor a perfect square trinomial the easy way. Now what about difference of squares? How can we factor those things?
Well, if you see something that looks like this, a squared minus b squared, it's just going to be a plus b times a minus b. That's the formula. So let's say if we have x squared minus 25. X squared is a perfect square because you can take the square root of it.
25 is a perfect square. Five times five is 25. So what you need to do is take the square root of x squared, which is x, the square root of 25, which is five, and one of them is going to be plus and the other is going to be minus. So knowing that, try this one.
X squared minus 36. and 4x squared minus 9 and 25x squared minus 81. So feel free to pause the video as you work on these. The square root of x squared is x. The square root of 36 is 6. And one of them is going to be positive and the other will be negative.
And that's all you got to do. Now for the next example, the square root of 4x squared is 2x. The square root of 9 is 3. So it's going to be 2x plus 3 and 2x minus 3. The square root of 25x squared is 5x.
The square root of 81 is 9. So the answer is 5x plus 9 and 5x minus 9. Now let's work on some more examples. Now what about these two? 3x squared minus 48 and 5x squared minus 45. Notice that we can't square root 3 or 48. It's not a perfect square. It won't give us an integer or a whole number. So what we got to do first is take out the GCF, which is 3. 3x squared divided by 3 is x squared.
Negative 48 divided by 3 is negative 16. Now we have a difference. perfect squares the square root of x squared is x the square root of 16 is 4 and so it's going to be x plus 4 times x minus 4 over 3 on the outside now for the next one we could take out the GCF as well, which is 5. And so we'll be left with x squared minus 9. The square root of x squared is x. The square root of 9 is 3. So it's going to be x plus 3 times x minus 3 times 5. Now what about a bigger example, like 16x to the 4th minus 81y to the 8th? What can we do here?
The square root of 16 is 4. The square root of x to the 4th is x squared. Simply divide the exponent by 2. The square root of 81 is 9. And the square root of y to the 8 is y to the 4th. Just divide the exponent by 2. Now, notice that we can factor this expression.
It's a difference of perfect squares. We can't factor the other one because it's a sum of perfect squares. Only a difference of squares can we factor. The square root of 4x squared is 2x. And the square root of 9y to the 4th is 3y squared.
And so, this is the answer. Now, in this video, we're going to talk about how to factor sums and difference of cubes. So, you need to know the formulas. So, let's talk about the sum of cubes. a cubed plus b cubed.
To factor it, it's going to be a plus b. times a squared minus a B plus B squared So while we have the equation on the board, let's work on an example So let's say if we have X cubed plus 8 So what that means is that? a to the third is equal to x cubed.
If we take the cube root of both sides, a is equal to x. Now b to the third correlates to 8. The cube root of a is 2, so b is 2. So now it's going to be a plus b, or x plus 2. a squared, that's going to be x times x, which is x squared. ab, that's x times 2, which is 2x. b squared, that's 2 squared, which is 4. And that's how you can factor it, using the formula. Now, let's try another example using the difference of squares.
a cubed minus b cubed. So this is going to be a minus b times a squared plus 2ab. plus b squared. Let's try two examples.
Let's try 8x cubed minus 27 first. So what is the cube root of 8? The cube root of 8 is 2, because 2 times 2 times 2, 3 times is 8, and the cube root of x cubed is x. The cube root of 27 is 3, so a is 2x, b is 3. a squared, that's 2x times 2x, that's 4x squared. Oh, this is not supposed to be 2ab.
This is supposed to be just ab. So let's just get rid of that. Now, ab is 2x times 3, so that's going to be 6x. And make sure to change the sign. If you see a negative sign here, it's going to be positive on the right side.
Now, b squared, that's 3 squared, which is 3 times 3, that's 9. So that's the answer. Let's try this example. X cubed minus 1 over 8. The cube root of x cubed is x. The cube root of 8 is 2, so the cube root of 1 over 8 is 1 over 2. a squared is going to be x squared. a times b, x times a half, that's 1 half x.
And then 1 half squared, that's 1 squared is 1, 2 squared is 4, so it's 1 fourth. And that's the answer. Let's try an example with large numbers.
125 x to the 6th minus 64 y to the 9th. So the cube root of 125 is 5. The cube root of x to the 6th is x squared. Divide the exponent by 3. The cube root of 64 is 4. And the cube root of y to the 9th is y cubed.
9 divided by 3 is 3. Now, a squared that's going to be 5x squared times 5x squared which is 25 x to the fourth and then change the sign plus 5 times 4 is 20 and then x squared times y cube just write them together and then plus b squared b is 4y cube 4 times 4 is 16 y cube times y cube is y to the 6 you got to add the exponents 3 plus 3 is 6 Now let's try another one with or using the sum of cubes equation. It's always good to write the equation first so you have something to follow along with. Try this one.
216x to the 12th power plus 343y to the 15th power. The cube root of 216 is 6. The cube root of x to the 12 is x to the 4. 12 divided by 3 is 4. The cube root of 343 is 7. And y to the 15 is going to be... y to the fifth power because 15 divided by 3 is 5 a squared that's going to be 6 times 6 which is 36 X to the fourth times X to the fourth you add 4 plus 4 and get 8 8 times B 6 times 7 is 42 and then it's going to be X to the fourth Y to the fifth you just multiply these two together And b squared, 7 squared, 7 times 7 is 49. Y to the 5th times y to the 5th, that's 5 plus 5 is 10, so this is the answer.
And now you know how to factor sums and differences of cubes. Now, what if you were to see a problem that looks like this? a squared plus 6a plus 9 minus b squared plus 8b minus 16. What would you do in this problem?
The first thing I would do is take out a negative 1 in the last three terms. So it's going to be b squared minus 8b plus 16. Now notice that we have a perfect square trinomial on the left and on the right. The first three terms represent a perfect square trinomial.
The square root of 1 is 1, the square root of 9 is 3. 1 times 3 is 3, times 2 will give us the middle term. And on the right side... The square root of 16 is 4. Times 1, that's going to be 4. Times by 2, you're going to get 8. Even though the sign is different, it's negative 8, it can still work. So now let's go ahead and factor it. The square root of a squared is a.
The square root of 9 is a. minus 3, so it's going to be a plus 3 squared, and then minus. On the other side, the square root of b squared is b, the square root of 16 is 4, but because of the negative sign, it's not going to be b plus 4, but it's going to be b minus 4 squared.
So now we have a difference of perfect squares. So using the formula a squared minus b squared is equal to a plus b times a minus b. We could factor it further.
So a is basically a plus 3, b is like b minus 4. So it's going to be a plus 3 and then plus b minus 4 and then a plus 3 minus b minus 4. So on the left, let's add. So it's going to be... a plus b and we can combine 3 and negative 4 3 plus negative 4 is negative 1 on the right we need to subtract it's going to be a minus b if you distribute the negative sign And then this is plus 3. And then negative negative b is plus 4. So 3 plus 4 is 7. So this is the answer.
a plus b minus 1. And a minus b plus 7. Now let's try another example. 4x squared plus 20x plus 25 minus 9y squared. minus 24y minus 16. So feel free to pause the video and work on this example. Now the first thing I'm going to do is I'm going to take out the negative 1 from the first three terms on the right. So it's going to be positive 9y squared plus 24y.
y plus 16. Now the first three represent a perfect square trinomial. The square root of 4 is 2, the square root of 25 is 5, 2 times 5 is 10, which is half of 20. So to factor the square root of 4x squared is 2x, the square root of 25 is 5, and just add an exponent of 2. Now the next one is a perfect square trinomial. The square root of 9 is 3, the square root of 16 is 4, 3 times 4 is 12, 12 times 2 is 24. The square root of 9y squared is 3y, the square root of 16 is 4, and so this is what we're going to have.
Now we have a difference of perfect squares. So it's going to be 2x plus 5 on both sides, and 3y plus 4. So here it's going to be positive, and here it's going to be negative. Now, on the left, let's add 2x plus 3y, 5 plus 4 is 9. On the right, don't forget to distribute the negative sign.
So it's going to be 2x minus 3y. 5 minus 4 is plus 1. And so that's going to be the answer. In this lesson, we're going to talk about how to solve equations by factoring. So let's start with a simple example.
6x squared minus... 30x is equal to 0. Find the value of x. So here we have a binomial. All we can do is take out the GCF. The GCF is 6x.
6x squared divided by 6x is x. Negative 30x divided by 6x is negative 5. So now we can use the zero product property rule. Zero times anything is zero. Therefore, if 6x is equal to zero, the whole thing is zero.
Or if x minus 5 is equal to zero, this equation will be true. Now solving for x, we can see that x is equal to zero. Zero divided by anything is zero. And in the other equation, if we add 5, we can see that x is equal to 5. So that's how you can solve an equation by factoring.
Let's try some more examples. Now what about this one, 3x squared minus 27? What is the value of x? Notice that we can take out the GCF, which is 3, and we'll be left with x squared minus 9. So we have a difference of perfect squares.
So to factor it, the square root of x squared is x, the square root of 9 is 3, and then we could set each factor equal to 0. So x plus 3 is equal to 0, and x minus 3 is equal to 0. So x is equal to negative 3 and positive 3. So that's the solution for this one. let's try one more example x squared minus 5x minus 36 so here we have a leading trinomial with a coefficient of 1 what two numbers multiply to negative 36 but add to negative 5 what would you say If we divide it by 2, we'll get negative 18. If we divide it by 3, we'll get negative 12. If we divide it by 4, we'll get negative 9. 4 and negative 9, they differ by 5. 4 plus negative 9 is negative 5. So to factor it... So, x is equal to negative 4 and positive 9. So now you know how to solve equations by factoring.