 Hi, this is a video about factoring polynomials. We will begin with the greatest common factor. First, the definition of factoring is we're writing a polynomial as a product of factors, things that are multiplied together. The greatest common factor of a list of integers is the largest integer or number that is a factor of all integers in the list. That means everything in the list must be divisible by this greatest common factor. To find the GCF of a list of integers, we can actually write each number as a product of primes, identify the common prime factors, multiply the common prime factors together to find the greatest common factor. As a little bit of a hint, if the bigger number is divisible by the smaller number, then the smaller number is the GCF. Find the greatest common factor of each. So I have 28 and 4. Let's do this to basic by writing each number as a product of primes. 28 can be written as 4 times 7. 7 is prime. 4 can be further broken down into 2 and 2. Each of these is prime. Prime just means the number is divisible by 1 and itself. 4 can be broken up into 2 and 2. Both of these are prime. So 28, the prime factorization, is 7 times 2 times 2. And 4, the prime factorization, is 2 times 2. This means that the common prime factors would be 1 copy of 2, 2 copies of 2. This means the GCF would have to be 2 times 2. The GCF is 4. Notice the easy thing to do here is out of 28 and 4, 4 is the smaller number, 28 is divisible by 4, meaning 4 is the GCF. And part B, 4 and 32, 4 is the smaller number. Is 32 divisible by 4? Yes. So 4 would have to be the GCF. Part C, 15, 18, and 66, the smallest number here is 15. Is 66 divisible by 15? No. So the best thing to do now would be to list factors of 15. Everything 15 is divisible by, including itself, and go through and say, OK, we know everything's not divisible by 15 in the given list. Everything's not divisible by 5 in the given list. However, 66, 18, and 15 are all divisible by 3. The GCF here, the highest number that each of the numbers in the list is divisible by would be 3. The greatest common factor is 3. When we're dealing with a list of variables, we have to make sure the variable is included in every term. And then we will pick that variable raised to the smallest power present. So in part A of example 2, we look and we see x is in every term. The smallest power would be 2. So the GCF would have to be x squared. Then furthermore, we go through the part B and we see we have y in every term. The smallest power given would be 1. So the GCF would have to be y to the first or just y. Example 3, we throw numbers and variables n together. So we'll have to find the number GCF, then we'll have to find the variable GCF to find the overall GCF. So what we have is we have 4, 12, and 16. 4, 12, and 16 are all divisible by 4. x squared, x to the fourth, x to the tenth. x is present in every term. The lowest power is 2. So x squared would be the variable GCF. This gives me 4x squared. Next up, we have 3 and 9. So the numeric GCF, 3 and 9 are both divisible by 3. y squared and y to the first, y is present in both terms. So y is the variable GCF. And then there's also x's in both terms. x to the fourth, x squared. So x squared would be the GCF for the x's. So 3x squared y would be your answer. So keep in mind, the variable has to be present in every single term. Now we'll actually factor out the greatest common factor out of a polynomial. To do this, we will take each term and divide it by the GCF. So we'll start off in part A with 10x squared plus 15x. The GCF is, well, 10 and 15 are both divisible by 5. That's the biggest number they're divisible by. x squared and x to the first. x to the first is the lower power. So what I have to do now is I have to take 5x and write it out front of a set of parentheses. And within the parentheses, I divide every term by the GCF of 5x. So I went through and I divided every term by 5x because that's what I'm pulling out front as the GCF. Simplifying inside the parentheses, I will get 10 divided by 5 is 2. x squared divided by x to the first would leave you with just x. Plus 15x over 5x would be 3. The GCF is now factored out. We can check to make sure we did this correctly. If I was to go through and redistribute the 5x to each term and parentheses, I should get the original question that I started with. 5x times 2x is 10x squared. And then 5x times 3 is plus 15x. This is just a way of checking to make sure you factored out the greatest common factor correctly. Now in part B, the GCF is going to be 3x to the sixth. So I have to take every term and I'm going to divide it by 3x to the sixth. So we write out every single term in parentheses, write 3x to the sixth underneath of it each time. Then we'll go in and we'll simplify. OK, 3 over 3 cancels out x to the eighth over x to the sixth. 8 minus 6 is 2. So that's x squared. 9 divided by 3, you're left with plus 3x. 7 minus 6 is 1. 18 divided by 3, that's minus 6. And x to the sixth over x to the sixth cancels out. We have our final answer. In part C, 6, 7, and 3 are divisible by nothing other than 1. So there's no numeric GCF. Furthermore, there's no B in every term, which means there's no GCF at all. There's no greatest common factor in part C, because the highest number that divides into all of them is 1 and B is not present in every term. All right, continuing on. In part D, let's look for the GCF first, and we'll factor it out. 8 and 40, the highest number there to be divisible by is 8. There's not x in both terms, but there is y in both terms. y to the fifth, because that's the smaller power. y to the fifth, y to the eighth. So you use the smaller power for GCF. So I'm going to take 8y to the fifth, write it out front of a set of parentheses, and I will take 8xy to the fifth and divide it by the greatest common factor. I will take the 40y to the eighth, and I will divide it by the greatest common factor. So we simplify here. The eighths cancel, the y to the fifths cancel, leaving you with x minus 40 over 8 is 5. 8 minus 5 is 3, so y to the third. That's your final answer with the GCF factored out. Now, in part e, it looks a little bit different, because now on the left half of the given question, we see we have a factor of a minus b, and then on the right half of the question, we see we have a minus b again. This is called a common binomial factor. It's a GCF that is the binomial a plus b. So literally what we're going to do now is take each of the terms and factor out at a plus b. We will divide by a plus b. So notice, okay, the a plus b's cancel out in the first fraction, and then the a plus b's cancel out in the second fraction. So as a shortcut, when you're dealing with common binomial factors, the common binomial factor will go in one set of parentheses in the final answer, and then the coefficients of the common binomial factor, three m squared n, and then the minus one will go in the second set of parentheses. So that being said, when we look at part f, you see you have a common binomial factor of x plus three. That means x plus three will go in one set of parentheses, and then the other set will be the coefficients, the five and the plus y, so five plus y. And that will actually lead us into factor by grouping, which we will cover next time. Thanks for watching.