In this video, we're going to talk about how to factor polynomials by taking out the GCF, by using difference of perfect squares, sum of perfect cubes, difference of cubes, factoring trinomials using substitution, factoring by grouping, completing the square using synthetic division, and also we're going to go over some difficult problems at the end. So let's start with the basics. So let's say if you have this binomial. 7x plus 21. How would you factor this expression?
Now the first thing you should always look for is the GCF, the greatest common factor. The greatest common factor between 7 and 21 is 7 because 7 goes into itself in 21. So you want to factor out a 7. 7x divided by 7, if we write it out. is x, so that goes here. 21y divided by this 7 gives you 3y, so that goes here.
And that's how you factor it by removing the GCF. So let's try some other examples. So go ahead and remove the GCF from 8x squared plus 12xy squared, and also from this one as well. 36x cubed y squared minus 60x to the fourth y cubed. So what number goes into 8 and 12?
So we can only remove one from each term. And the same is true for y. We can remove 1y from each term.
So if we take this term, 8x squared, and divide it by our GCF, 4xy, we're going to get 2x. And if we divide 12xy squared by 4xy, that's going to give us 3y. So that's what's left over. To show you work, you can do this. 12xy squared divided by 4xy.
Notice that the x is cancelled. 12 divided by 4 is 3. y squared over y, you've got to subtract the exponents. 2 minus 1, you get y to the 1. And that's what we put here.
So that's how we can factor that binomial by removing the GCF. So let's try this one. What is the greatest common factor between 36 and 60?
12 goes into 36 and 60. And we can take out at least 3x variables from both terms. And we can take out two y variables from each term. So if we take this term and divide it by 12x cubed y squared, 36 divided by 12 is 3. And we took out both, all three x's in all two y variables.
So there's nothing left over. Now for the next term, negative 60 divided by 12 is negative 5. And out of the four x variables, we remove 3. So there's one left over. And out of the three y variables, we only took out two, so there's one y left over. And that's how you factor it. So, how would you factor this expression, x squared minus 25?
Notice that you can take the square root of x squared and 25, so this is a difference of perfect squares problem. And the equation that you need... a squared minus b squared when you factor it's going to be a plus b times a minus b so basically what you got to do is take the square root of x squared the square root of x squared is just x the square root of 25 is 5 and on one side you're going to have a plus and on the other side a minus So using that, go ahead and factor y squared minus 64 and 8x squared minus 18. Feel free to pause the video as you try these examples.
So the square root of y squared is just y. And the square root of 64 is 8. And it's going to be plus and minus. Now, we can't square root 8 or 18. But we can take out the GCF.
We can factor out a 2. 8x squared divided by 2 is 4x squared. And 18 divided by 2 is 9. Notice that 4 and 9 are perfect squares. So now we can apply this equation. So we're going to keep the 2 on the outside. The square root of 4x squared is 2x.
And the square root of 9 is 3. And on one side we're going to have a positive side, and on the other side a negative. So that's the answer for that problem. go ahead and try these examples 81 x squared minus 36 y squared and try this one as well 200 x to the fourth minus 200 88 y to the 6. So we can take the square root of 81 and 36. Those are perfect squares.
So we can factor this one directly. The square root of 81 x squared is 9x, and the square root of 36y squared is 6y. So we're going to have a plus, and we're going to have a minus.
Now notice that we can factor further. Notice that 9 and 6 has a GCF, which is 3. So in this term, we can take out a... 3 if we do so it's going to be 3x plus 2y and we could take out a 3 from here as well and it's going to be 2x I mean 3x minus 2y so we can multiply by these two numbers so we have 9 times 3x plus 2y and 3x minus 2y the fact that we have a 9 means that we could remove a GCF let's say if we took out a 9 we would have 4x squared not 4x squared excuse me 9x squared 9x squared minus 4y squared, which would eventually factor to this expression. So, even if you don't take out the GCF at the beginning, you can take it out at the end, and you can still get the same answer. The square root of...
9x squared is 3x and the square root of 4y squared is 2y. So both ways can work. In the next problem, we're going to factor out a 2. If we take out a 2, we're left with 100, which is a perfect square, x to the 4 minus 144, y to the 6. So let's go ahead and factor it using the difference of perfect squares equation.
So the square root of 100, x to the 4th is, well the square root of 100 is 10, and the square root of x to the 4th, you just divide the 4 by 2, you get x squared. The square root of 144 is 12, and the square root of y to the 6th, divide the 6th by 2, you get 3. So, 12y cubed. And on one side, it's plus, the other side is minus.
Now, 10 and 12 are both even numbers, so we can take out a 2. And if we do so, we'll be left with 5x squared plus 6y cubed. And we can take out another 2 from this term. and so we'll have 5x squared minus 6y cubed.
So if we collect these three numbers on the outside, it's going to be, let's see if I can fit it here, 8 times 5x squared plus 6y cubed times 5x squared minus 6y cubed. And that's the answer for this problem. So now, how would you factor this trinomial? x squared plus 11x plus 30. How would you do it?
Now notice there's a 1 in front of the x squared. The leading coefficient is 1. When asked the case, all you need to do is look at the last term. 30 and find two numbers that multiply to 30 but add to 11 let's start with 1 30 divided by 1 is 30 30 divided by 2 is 15 30 divided by 3 is 10 30 divided by 5 is 6. 4 doesn't go into 30. Now we need to pick a pair of numbers that add up to 11. 5 plus 6 is 11. So the factor is just going to be x plus 5 times x plus 6. By the way, if you need to solve the equation, let's say this was equal to 0, and you want to find the zeros, set each factor equal to 0 and solve for x. So here x would equal negative 5 and x would equal negative 6. That's how you can find the zeros. of a polynomial by factoring so let's try this one x squared plus 2x minus 15 so what are two numbers that most Supply the negative 15, but add to the middle term 2. So let's divide negative 15 by 1. We'll get negative 15. It doesn't go into 2, but if we divide it by 3, we get negative 5. And if we divide it by 5, we get negative 3. Notice that 5 plus negative 3 is equal to positive 2. So the factor, it's just x plus 5 times x minus 3. Now what about this problem?
x squared minus 2x minus 48. Let's look for two numbers that multiply to negative 48, but add to the middle term negative 2. So if we divide negative 48 by 1, we're going to get... negative 48 if we divide it by 2 it's going to give us negative 24 if we divide it by 3 negative 16 by 4 negative 12 5 doesn't go into negative 48 if we divide it by 6 negative 8 if we divide it by 8 negative At that point, the numbers will repeat in reverse order. So we need two numbers that add up to negative 2. 6 plus negative 8 is negative 2. So the answer is x plus 6 and x minus 8. If it was equal to 0, the solution to the equation would be negative 6 and positive 8. You've got to change the sign. So let's try this one, x squared minus 9x plus 20. So let's look for two numbers that multiply to 20 but add to negative 9. If we divide 20 by 1, we're going to get 20. If we divide it by 2, we'll get 10. 3 doesn't add to 20. going to 20 but if we divided by 4 will get 5 now 4 times 5 is 20 before plus 5 adds up to 9 negative 4 and negative 5 also multiplies to positive 20 but they add to negative 9 so the answer is X minus 4 times X minus 5 so that's how you can factor a trinomial when the leading coefficient is 1 now what can we do when the leading coefficient is not 1 so let's say if you want to factor 2x squared minus 5x minus 3 now notice that the leading coefficient is a small number it's 2 so sometimes it might be easier to factor it by trial and error other times you can use the method Some people call it the AC method, but for this one, let's factor it by trial and error, because the numbers are small.
So, we need two numbers that multiply to 2x squared. The only way to make this work is 2x and x. It has to be that way.
Now, we need two numbers that multiply to negative 3. So, to get 3 in the first place, we need a 1 and a 3. The question is, should it be like this, 1, 3, or should it be 3? one right now is let's try 31 if we multiply 2x by one will get 2x and 3 times x is 3x now the only way can add up to negative 5x is if we have negative 2x and negative 3x however to get negative 3 we need a positive and a negative number so that's not going to work So, we're going to change the order. We're going to put the 3 here and the 1 here. 2 times 3 is 6x.
1 times x is 1x. Now, 6 plus 1 is 7, but 6 minus 1 is 5. Now, this could work. Since you want a negative 5, we need a negative 3 and a plus 1, because 2x times negative 3 is negative 6x, and 1 plus x is 1x, that adds to negative 5x.
And that's how you can factor it using trial and error. Now, let's say if you don't like that method. If you want a consistent method to follow, here's what you could do.
Let's use the same equation. We're going to multiply the first and the last coefficient together. So 2 times negative 3 is negative 6. And we need two numbers that multiply to negative 6 but add to the middle term, negative 5. So if we divide it by 1, we're going to get negative 6. If we divide it by 2, we'll get negative 3. 1 plus negative 6. 6 is negative 5 so we're going to replace negative 5x with positive 1x and negative 6x because X minus 6s is negative 5x In the first two terms, we're going to factor out the GCF.
The greatest common factor between 2x squared and x is just x. 2x squared divided by x is 2x, and 1x divided by x is 1. Now, we're going to take out the GCF between the last two terms. We're going to factor out negative 3. Negative 6x divided by negative 3 is 2x. Negative 3 divided by negative 3 is plus 1. If these two are the same, then you're on the right track. So we're going to factor out 2x plus 1. Now this whole thing is one term.
If we take out the 2x plus 1, we're left with x, which goes here. And if we take out the 2x plus 1 from this term, we're left with negative 3, which goes here. And that's our answer, 2x plus 1 and x minus 3. so let's try some more examples using that method so feel free to try this problem 3x squared plus 5x minus 2 so let's multiply the first and last coefficient so 3 times negative 2 is negative 6 and if we divide negative 6 by 1 we're going to get negative 6 if we divide it by 2 will get negative 3 if we divide it by 3 will get negative 2 and if we divide it by 6 will get negative 1 so which pair of numbers add up to 5 6 plus negative 1 is equal to 5 so what we're going to do is replace 6 I mean 5x with 6x minus 1x And now let's factor by group.
So between 3x squared and 6x, the GCF is 3x. 3x squared divided by 3x is x. 6x divided by 3x is 2. Now for the last two terms, we're going to...
take out a negative 1 negative 1x divided by negative 1 is X negative 2 divided by negative 1 is plus 2 so here we have a common factor we're going to factor out X plus 2 so if we take out X plus 2 we're left with 3x and if we remove x plus 2 here we're left with negative 1 and that's how we factor it so let's try one final example on this topic let's try 6x squared minus 7x minus 3 so let's multiply 6 and negative 3 6 times negative 3 is negative 18 and if we divide it by 1 we're going to get negative 18 if we divide it by 2 we'll get negative 9 2 plus negative 9 adds up to the middle term negative 7 so we have a 6x squared plus 2x minus 9x minus 3 in the first two terms we're going to take out the GCF which is 2x 6x squared divided by 2x is 3x 2x divided by 2x is 1 and the last two terms we're going to take out a negative 3 negative 9x divided by negative 3 is negative 3x and negative 3 divided by negative 3 is plus 1 so here's our common factor we're going to factor it out and if we take out 3x plus 1 here we're left with 2x and if we remove 3x plus 1 from the second term we're left with negative 3 and so that's how we can factor a trinomial when the leading coefficient is not 1 So our next topic, we're going to talk about how to factor polynomials with three terms by using substitution. So let's say if you have x to the fourth plus 7x squared plus 12. So notice that this exponent is twice the value of that one. When you see that, you can factor it just as any trinomial as we did before, where the leading coefficient is 1. So let's substitute it with a variable.
Let's say a is equal to whatever the middle term is, x squared, just without the coefficient. So therefore, a squared is x to the fourth. So we have...
a squared plus 7a plus 12 so we need two numbers that multiply to 12 add to 7 12 divided by 1 is 12 12 divided by 2 is 6 12 divided by 3 is 4 but 3 plus 4 is 7 so this is going to be a plus 3 times a plus 4 which is if we replace a with x squared it's x squared plus 3 and x squared plus 4 So that's the answer for that example. Let's try this one. X to the 6. actually hold on negative 2x to the 6 plus 6x cubed plus 56 now notice that we have even numbers here so we can do is take out the GCF which is negative 2 we left with x squared minus 3x cubed minus it's 28 so now we're going to focus on this part here but let's substitute the middle term with a so we're going to say a is X cubed which means that a squared is X to the 6 so what we now have is negative 2 a square minus 3a minus 28 so we need two numbers that multiply to negative 28 but adds to the middle term negative 3 so if we divide it by 1 we'll get negative 28 if we divide it by 2 we're going to get negative 14 3 doesn't go into 28, but 4 does. Negative 28 divided by 4 is negative 7, and 4 plus negative 7 is negative 3. So the factor is going to be a plus 4, and a minus 7. So we're going to replace a with x cubed, so it's going to be x cubed plus 4, and x cubed minus 7. try this one x to the negative 2 plus 13 x to the minus 1 plus 40 and this one we're going to solve it as well in addition to factoring this particular function So we're going to set a equal to the middle term, which is x to the minus 1. So right now we have a squared plus 13a plus 40. So we need two numbers that multiply to 40. But that adds to 13. 40 divided by 1 is 40. Divided by 2 is 20. Divided by 4 is 10. Divided by 5 is 8. And 8 plus 5 is 13. So it's A plus 5. plus 8 and at this point we can replace a with X to the minus 1 plus 5 and X to the minus 1 plus 8 So if we set each factor equal to 0, x to the negative 1 is equal to negative 5, and x to the minus 1 is equal to negative 8. Now to solve for x, you need to understand that x to the negative 1 is the same as 1 over x. So 1 over x...
is equal to negative 5 which we can write it as negative 5 over 1 and we can cross multiply so 1 is equal to negative 5x dividing both sides by negative 5 our answer is negative 1 over 5 so for the other one if we follow the same process X will be negative 1 over 8 so that's how you solve a particular function like that So what about this one? Let's say if e to the 2x, if we have e to the 2x plus 10 e to the x plus 21, how would you factor it? Notice that the exponent here is twice the value of the middle one.
So this is another substitution problem. So we're going to set a equal to this middle term, e to the x. So a squared would be e to the 2x.
So what two numbers multiply to 21 but add to the middle term 10? 10 this would have to be 3 and 7 3 times 7 is 21 but 3 plus 7 is 10 so right now what we have is a squared plus 10 a plus 21 and when we factor it's going to be a plus 3 times a plus 7 and then you need to replace a with e to the X so it's e to the X plus 3 times e to the X plus 7 and that's how you factor so our next topic that you need to know is the sum and difference of perfect cubes so let's say if you want to factor X cubed plus 8 the equation that you need is a to the third plus the B to the third is equal to a plus B times a squared minus a B plus B squared So a to the third is x to the third, and b to the third is 8. So to find a, you need to take the cube root of x to the third. The cube root of x to the third is x.
You just divide the 3 by 3. cube root of a is 2 because 2 times 2 times 2 is 8 so a squared if a is if a is X a squared therefore is x squared and then we have minus a times B or X times 2 which is just 2x and B is 2 so B squared or 2 squared is 4 and that's how you factor it so let's say if you want to factor y cube minus 125 this is the difference of perfect cubes so if there's a minus here this is going to be minus and this is going to plus everything else is the same the cube root of y cube is just y and the cube root of 125 is 5 because 5 times 5 times 5 3 times is 125 and because we have a minus sign it's going to stay negative see these two signs stay the same but it switches here so a squared is y squared and to find a B is going to be plus we change the sign from negative to positive plus 5 times y which is just 5y and then be squared 5 squared is 25 Let's try another example. So let's say if we want to factor 27x cubed plus 64y to the 6th. So the cube root of 27 is 3, and the cube root of x is 3. x to the third is x.
The cube root of 64, since we have a plus, this is going to be plus. The cube root of 64 is 4. 4 times 4 times 4 is 64. And the cube root of y to the sixth, you just got to divide this by 3, is going to be y squared. so keep in mind this is a and this is B so a squared 3x times 3x will therefore be 9x squared and then minus a be 3x times 4y squared is 12 X Y squared and then B squared so 4y squared times 4y squared is 16 Y to the fourth and that's how you factor it you just got to follow this equation you can solve for a and B and just substitute it in the equation So now let's say if we have this cubic polynomial and we want to factor by grouping.
Now we already use this method when we factored using the AC method when the trinomial had a leading coefficient that wasn't 1. But here's something you need to know. Notice that the first two coefficients, 1 and 2, has the same ratio as negative 5 and negative 10. 2 divided by 1 is 2. Negative 10 divided by negative 5 is 2. When the first two coefficients have the same ratio as the last two coefficients, coefficients you can therefore factor by group if that ratio doesn't if the ratio is not the same it won't work so in the first two terms let's take out the GCF which is x squared and so x cubed divided by x squared is x 2x squared divided by x squared is 2 and in the last two terms let's take out a negative 5 so we'll be left with x plus 2 so here's our common factor we're going to take out x plus 2 and then we'll be left with x squared minus 5 and that's how you do it let's try another one so 4x cube minus 8x squared plus 3x minus 6 so negative 8 divided by 4 is negative 2 and negative 6 divided by 3 is also negative 2 So they have the same ratios, which means we can factor by grouping. In the first two terms, let's take out the GCF, which is 4x squared.
And so we'll be left with x minus 2. And in the last two terms, let's take out a 2. Actually, not a 2, but a 3. And so we'll also have x minus 2. So our answer is going to be x minus 2 times 4x squared plus 3. So, how would you factor x squared minus 4 and x squared plus 4? How would you do it? Now, we covered the first one already. That's the difference of perfect squares. We know it's going to be x plus 2 and x minus 2. But what about x squared plus 4?
plus 4, a sum of perfect squares. Can we factor that? The answer is no, we can't factor it using real numbers.
However, we could factor it using imaginary numbers. x squared plus 4 is x plus 2i. and x minus 2i. i is equal to the square root of negative 1. If you haven't learned imaginary numbers, you can skip this section and move on to the next topic. But I'm going to foil it.
x times x is x squared. x times negative 2i is negative 2ix. 2i times x is plus 2ix. And 2i times negative 2i is negative 4i squared.
So the two middle terms cancel, and you get x squared minus 4i squared. now notice that I is the square root of negative 1 I squared is equal to negative 1 so you now have this x squared minus 4 times times negative 1, which is x squared plus 4. So that's how you can factor the sum of perfect cubes, using imaginary numbers. So let's try some examples, including the use of radicals.
Let's say if you want to factor x squared plus 9, x squared minus 3, and x squared plus 3. So when you have the sum of perfect cubes, You're going to use imaginary numbers, but I'm going to compare it to x squared minus 9 x squared minus 9 is going to be x plus 3x minus 3 But x squared plus 9 it's going to be x plus 3i and x minus 3i. So all you have to do is add an i to it. Now if you want to factor x squared minus 3, here's what you need to do. It's the difference of perfect squares. So the square root of x squared is and the square root of 3 is actually radical 3 so we have plus and minus now here we have x squared plus 3 so it's just x plus root 3 times I and x minus root 3 times I so depending on what level of course you're taking you may or may not need to know that so now we're going to focus on factoring higher degree polynomials using synthetic division so let's say if you have x cubed minus 2x squared minus 5x plus 6. Notice that 1 and 2 doesn't have the same ratio as 5 and 6, so we can't factor by grouping.
So we need a different method. So what you want to do is make a list of your possible factors. Possible factors factors of six are 1, 2, 3, and 6. And possible factors of the leading coefficient 1 is just plus or minus 1. So all of these are possible answers.
Let's start with positive 1. Let's see if that works. So we're going to use synthetic division. So the coefficients that you see, 1, negative 2, negative 5, and 6. We're going to place them here and bring down the 1. In synthetic division, you're going to multiply, add, multiply, add, and just repeat the process.
So, 1 times 1 is 1, and then negative 2 plus 1 is negative 1, and then this 1 times negative 1 is negative 1. Negative 5 plus negative 1 is negative 6. 1 times negative 6 is negative 6. 6 to negative 6 is 0. So, usually when you use synthetic division, this goes down by 1 level. level if this was X to the third this one's going to be x squared so we have 1x squared minus 1x minus 6 now we can use synthetic division again but right now we have a trinomial we know how to factor trinomial so two numbers that multiply the negative six but add to the middle term negative one would be let's see we have one in negative six two and negative three and this is what we're looking for so this trinomial can be factored to x plus 2 times x minus 3. However, our first solution was positive 1. In its factored form, it is x minus 1. So this right here is how you factor the original expression. If you multiply these three factors, you're going to get this original equation. By the way, if you get a 0, that means this number works. It's a factor.
If this wasn't a 0, you have to pick a different number. These are the... possible factors as you can see 2 negative 3 and negative 1 they're all in possibilities here let's try another one like this so let's say if we have X to the fourth plus 2x cubed plus x squared plus 8x minus 12 so possible fact of 12 are 1 2 3 4 6 and I'm out of space but 12 as well and possible factors of the lean coefficient 1 is just 1 So, let's see what numbers are going to work for this one. Let's try 1 again. Let's see what happens.
So, let's put the coefficients. 1, 2, 1. 8 negative 12 by the way if one of these variables were missing let's say if there's if there was no x squared if it was like 2x cubed plus 8x make sure you put a 0 in place for the x squared don't forget that that's important so let's bring down the 1 1 times 1 is 1 2 plus 1 is 3 1 times 3 is 3 1 plus 2 is 4 1 times 4 is 4 8 plus 4 is 12 1 times 12 is 12 those two adds to 0 so right now we have this was 4 so now it's going to go down to 3 1x cubed plus 3x squared plus 4x plus 12 now notice that the coefficients 1 and 3 have the same ratio as 4 and 12 12 divided by 4 is 3 3 divided by 1 is 3 which means we can factor by grouping so in the first two terms Let's take out the GCF, we know it's x squared, and we'll be left with x plus 3. And the last two terms, let's factor out a 4, and we're going to get x plus 3. So the answer is x squared plus 4 times x plus 3. But that's just for this part. Now keep in mind we have a 0 here, so its factored form is x minus 1. So this is how we can completely factor the problem using real factors. Now, if you want to include imaginary numbers, you can factor the x squared plus 4, which is the sum of perfect squares. And if you want to factor, it's going to be x plus 2i, x minus 2i, and then times x plus 3. So the next thing we're going to do is factor trinomials by completing the square.
So let's say if you have x squared plus 6x plus 7. Now we're going to ignore the 7. at least for now look at the 6x look at the coefficient 6 take half of 6 half of 6 is 3 and then square that we still have the plus 7 since we added 9 we change the value of the expression which is not good So to balance that, we need to take away 9. So if we add 9 and subtract 9, we haven't changed the value of the expression. And that's what we want to do. Now this term is x squared plus 6x plus 9. Two numbers that multiply to 9 but add to 6 is 3 and 3. So it can be factored as x plus 3, x plus 3. This is a perfect square because the 3s are the same.
But you really don't need to use this method. There's a shortcut in which you can use to factor that perfect square. And here's what you need to do.
So whatever this variable is, it's going to go here, x. Whatever this sign is, it's going to go there, plus. And whatever this number is before you square it, so not 9 but 3, it's going to go here. And this 2 is going to go here as well. And thus this can be written as x plus 3 squared because there's two of them.
And then these two numbers. we simply need to combine 7 minus 9 is negative 2 and that's how you can factor by completing the square so let's try another example so let's say if we have x squared minus 4x plus Now that method is going to be extremely useful when you have fractions, because you don't want to factor a fraction. So let's take half of 4. Half of 4 is, ignore the negative sign, half of 4 is 2, and we're going to square it.
And since we're adding 2 squared, which has a value of 4, we need to subtract 4 so that the expression maintains its original value. So the factor of this expression is going to be x minus 2 squared. And 12 minus 4 is 8. That's it. Alright, let's try this example.
x squared minus 3x plus 1. So we have x squared minus 3x. Half of 3 is 3 over 2. And we gotta square it. So plus 1. Now 3 over 2 squared.
3 squared is 9. 2 squared is 4. Since we added 9 fourths, we have to take away 9 fourths. So the factor is gonna be x minus 3 over 2 squared now to combine 1 and 9 over 4 we need to get common denominators 4 over 4 is the same as 1 so 4 over 4 minus 9 over 4 is negative 5 over 4 so this is our final answer x minus 3 over 2 squared minus 5 fourths So let's try this one 2x squared plus 6x minus 9 so first we need to factor out a 2 so we'll be left with x squared plus 3x and half of 3 is 3 over 2 and we're going to square it and then we have minus 9 so we've added 3 over 2 squared so I'm going to take away 3 over 2 squared but notice that the 3 over 2 squared is distributed by this 2 so we need to incorporate that to there so we're going to leave the 2 on the outside and then it's going to be X plus 3 over 2 squared and then we have minus 9 minus 3 squared is 92 squared is 4 times 2 so this part is going to stay the same we don't have to worry about it anymore and that's a terrible 3 i did it again 9 over 4 times 2. If we divide 4 by 2 backwards, we get 2. So we're just going to get a 2 on the bottom. And now we need to get common denominators.
So we're going to multiply this by 2 over 2. So this is going to be negative 18 divided by 2 minus 9 over 2. And negative 18 minus 9 is negative 27 over 2. And we still have our original expression. So this is the answer. That's how we can factor by completing the square. So that's enough with that topic.
So now let's factor some advanced factoring polynomial problems. So let's say if you get x squared minus 2xy plus y squared minus 9. What would you do here? When there's so many variables, how would you factor this expression? Now feel free to pause the video and try it out yourself.
It helps to see if you focus on this part first. Now, you need to know some equations. a plus b squared is a plus b times a plus b, and when you FOIL it, it becomes a squared plus 2ab.
plus b squared. a minus b squared is a squared minus 2ab plus b squared. Notice that this equation is the same as what we see here.
Instead of a and b, it's just x and y. So you can factor those three as x minus y squared. And we still have a negative 9. Now notice that we have a difference of perfect squares. Like x squared minus 25, which when we factor it, we know it's x plus 5, x minus 5. So we have that same situation here. But I'm going to use brackets this time.
If we take the square root of x minus y squared, the square goes away. And we're just left with x minus 5. y in both parentheses I mean brackets and if we take the square root of 9 it's 3 so we have X minus y plus 3 X minus y minus 3 and that's how you factor Try this one. x squared y squared minus y squared minus z squared plus x squared z squared.
How would you factor that expression? Notice that the first two terms have a y squared, and the last two terms have a z squared. Now notice that we have a minus y squared here.
It's like positive, then negative, but here it's negative. negative then positive I'm going to rearrange it so that it's in the same order positive and then negative and then the second part I'm going to write it as x squared z squared minus z squared because notice that these two are similar And these two are similar. The coefficient is 1 and negative 1. 1 and negative 1. They have the same ratio, which means we can probably factor by group.
So in the first two terms, let's take out the GCF, which is y squared. So we're going to be left with x squared minus 1. And in the last two terms, let's take out the GCF, which is z squared. And we're going to be left with x squared minus 1. So we have a common factor, x squared minus 1. one if we take that out here we're going to have y squared and here we're going to have plus z squared so the factor x squared minus one is going to be x plus one times x minus one and then y squared plus z squared so that's the answer for this problem and if you want imaginary solutions I guess you can say y plus zi and y minus zi but I don't think that's necessary for this problem but that concludes this video so we covered almost every factoring technique shortcut trick that you'll probably need as you go through your algebra and trigonometry and pre calculus courses and even calculus you can take this information and it's going to help you along your way so thanks for watching this video and have a great day