We've seen how to factor a polynomial with factoring by grouping. But, sometimes, it's easier to just recognize special patterns to make factoring easier. We're going to look at those special patterns in this lesson. The special patterns involve the square of binomials and the difference of two squares. When we multiplied polynomials, we saw how to expend each of these three cases. But this time, we're going to be given these forms and we're going to move in the other direction. We want to find the factored form. We're going to focus on these two patterns first. We're going to factor the perfect square trinomials as a perfect square or a squared binomial.
See if you could find the pattern and figure out what a and b would be to factor this expression. Now, I know I haven't even shown you how to do this, but, I think you'll be able to figure it out. Write your factors with parentheses in this box. Good luck.
x plus 10 squared is correct. Great job if you got that one right. If you're a little stumped, then stay with me and we'll figure this out together. You might've also put x plus 10 times x plus 10, which is also correct. We have the same factor listed twice, so we could've just written it as x plus 10 squared. This expression follows this factoring pattern. The first term is a perfect square. Since x squared is x squared, and the last term is also a perfect square. Since 10 squared is 100. So the a is really x, and the b is really 10. This means that we can rewrite this expression as x squared plus 2 times x times binomial squared. So, we really have x plus 10 squared. Don't worry if you didn't get this first one right, there'll be more to come.
Here's a different trinomial, and I want you to try and figure out its factored form. Write your answer in this box.
Here the answer should be the quantity x minus 3 squared. Great pattern recognition if you figured that one out. We can see that the first and last terms are perfect squares, x squared is x squared and 3 squared is 9. This term is negative which fits my negative sign in the factoring pattern, so I'll have x squared minus 2x times 3 plus 3 squared. The a is really x and the b is really parentheses, x minus 3 squared. The one thing we want to notice is that, if this term is positive, then we use a positive in the squared binomial. If this term is negative, then we use a negative sign in our factored form.
What do you think would be the factored form for this expression?
The quantity 4x plus 5 squared is correct. Fantastic work if you got that one right. We can see that this expression fits this first factoring pattern. We have a perfect square trinomial. The first term is a perfect square, a 4x squared, and the last term is a perfect square, a 5 squared. So the a term is really 4x and the b term is really 5. So to write my factor form, I simply write
Now that we've looked at factoring perfect square trinomials, let's look at the difference of squares. If ever I have the difference of 2 squares, I can factor it as a plus b, times a minus b. This factoring pattern is really helpful, and we'll see it throughout the course. So let's focus on this difference of squares and see if you can factor a couple examples. Try factoring 49x squared times 4. What do you think are the two factors we need?
I know this is a perfect square since 7x squared is 49x squared. 4 is also a perfect square, since 2 squared equals 4. So, the a is really 7x, and the b is really positive 2. We can factor the difference of squares as a plus b times a minus b. So the two factors are 7x plus 2 and 7x minus 2. This is our answer. You could have also switched the order of the factors and you would still have gotten it right.
What do you think would be the factors for this expression? Be sure your answer is completely factored. That is, there should not be a common factor between any two terms inside of a parenthesis.
This question was a bit tricky. Whenever we start factoring, we should remove the greatest common factor first. 9 is the greatest factor that goes into 9x squared and 36y squared. So I can factor that 9 out of my expression and I'll be left with x squared minus 4y squared. So the first term is a perfect square since x squared equals x squared, and the second term is also a perfect square since 2y squared is 4y squared. We can factor this inside parenthesis using a+b times a-b. When I write the factored form, I want to remember to keep my GCF, my greatest common factor. That stays on the outside of the other parentheses. So here's our factored expression. And in this case we have three different factors, one of them being the GCF of these terms. You might have thought to factor this as 3x plus 6y times 3x minus 6y. There's a common factor of 3 in this term, so we can factor it out. There's also a common factor of 3 in these two terms, so we can factor a 3 out from this parentheses as well. So I've 3 times x plus 2y times 3 times x minus 2y. And we can easily see by distributing we'd wind up with these parentheses back from before. We know the order of multiplication doesn't matter, so I can just multiply the 3 times 3 and then list my other two factors. This would have been another way to get our same result. If you want to avoid removing common factors between pairs of terms, then it's best to factor out the GCF first. If we do that, then we won't need to do this in the end.
I hope you've gained some insights into factoring more quickly using these patterns. Before we get to some practice on this factoring, let's review our thinking. First we want to decide if there's a GCF. If there is, we should factor it out first and keep it on the outside. If there's not a GCF, then we should ask is there a special pattern here? The special patterns make our life a little bit easier and we can use these to identify the factors. If there's not a special pattern, we want to use factoring by grouping. We can pull out common factors from four terms and then pull out another common factor from those two terms.
Try factoring this expression. See if it fits one of the three factoring patterns we've covered and then identify the a and the b.
This perfect square trinomial is really a binomial squared. Our a is really 3c, since the quantity 3c squared equals 9c squared. Our b term is really 2d. We know 2d squared equals 4d squared. So this is a perfect square trinomial. And we can factor it as a minus b squared. So we have 3c minus 2d squared.
What would this be in factor form?
This perfect square trinomial fits this pattern. Our a is really 3x and our b is really 4y. So our final answer is 3x plus 4y squared. Great work on this one.
Try factoring this expression.
Whenever we start factoring, we want to first look for a GCF, a Greatest Common Factor. There is a Greatest Common Factor for these two terms, it's 3. So I can see that I have a GCF times the quantity a squared minus b squared. The a is really 3x and the b is really 2. So to factor this pattern, we use a plus b times a minus b. We want to make sure we include our GCF in the front, and then we have 3x plus 2, times 3x minus 2, our a plus b, and a minus b.
Try this one out.
As always we want to start by looking for our greatest common factor. And yes indeed, there is one. It's 5. We take up the 5, leaving us with x squared minus squared minus b squared. Our a is x and our b is 2y. So here's our final answer.
Try factoring this trinomial.
Here's the correct answer. Excellent work if you got it. When we start to factor, we want to look for a greatest common factor first. The three terms don't share an x, so I can't factor an x out. The three terms also don't share y, since this first term doesn't have one. And when we look at the coefficients, the greatest common factor is just one. This trinomial is a perfect square trinomial pattern. The a is to x and the b is 5y. So we can factor our trinomial using a minus b squared. 2x minus 5y squared.
Try factoring this one.
This would be the factored form. Amazing work if you got it right. This one was really tough. As always we want to look for a greatest common factor first. And it turns out eight goes into both terms. So I have 8 times one minus 4x squared. So we have a greatest common factor sitting on the outside, and a squared minus b squared sitting on the inside. The a's really 1, since the quantity a squared is 1. And the b is really 2x since this quantity squared is 4x squared. We keep our GCF on the outside and then we factor using our pattern, a plus b and a minus b.
Let's try another practice problem for factoring patterns. Here, we have 18a squared minus 84ab plus 98b squared. Factor carefully and put your answer in this box.
The answer's 2 times 3a minus 7b squared. This was an example of a perfect square practice problem. Did you get the right answer? Congratulations if you did. Let's see how we got that answer. Remember, whenever I factor any polynomial, I want to look to see if there's a greatest common factor. When I look at the variables, I can see there is not a variable portion of a greatest common factor. But, I notice 18, 84 and 98 are all even numbers. Since they're all even numbers, I know that there is a greatest common factor of two. So the first thing I want to do is factor out the two. When I factor out the greatest common factor of two. I'm left with 9a squared minus 42 ab plus 49b squared. What I notice when I look at this problem is that the first term and the last term are perfect squares. Let me write it that way. 9a squared can be written as the quantity 3a squared. 49b squared can be written as the quantity 7b squared. When I see a perfect square in the first term, and a perfect square in the second term. I have to check to see if the middle term is 2 times the square root of the first term times the square root of the last term. Let's check. 2 times 3a times 7b is equal to. 21 times 2 is 42 ab. So we know that this is a. Perfect square. Since the first term is 3a squared and the last term is 7b squared and the middle term is 2 times the square root of the first term times the square root of the last term, I know what's a perfect square. Now knowing that it's a perfect squire I have to remember the formula. The square root of the first term is 3A. I nee a minus because the middle term is minus. And then the square root of the last term which is 7B and we square it. Now a good thing to do when you're doing these factoring problems is to check your asnwer. When you check you answer, let's foil this out. 3 a minus 7 b squared is 3 a minus 7 b times 3 a minus 7 b. 3 a times 3 a is 9 a squared. 3 a times negative 7 b is negative 21 a b. Negative 7 b times 3 a is negative 21 a b. And negative 7 b times negative 7 b is Positive 49b squared. So I can combine the 2 middle terms. I can also multiply it through by 2. When I finish multiplying through by the gcf, I get 18a squared minus 84ab plus 98b squared. Which is what we started with. So we know that we got the right answer.