Let's see if we have the skills to solve our original equation. We've learned about the greatest common factor, how to factor using special patterns, and how to factor by grouping. Can we use any of those techniques on the left-hand side to help us solve this equation? Answer these questions using yes or no.
The answer to all three of these questions is no. The only factor common to all three terms is 1. We also know that we can't use a factoring pattern to factor this side of the equation. 6 is not a perfect square, so this doesn't fit the pattern for a perfect square trinomial, or for the difference of two squares. And finally, we need four terms to do factoring by grouping so this won't work either.
Let's look at the left size of that equation more closely. We could remove an x from the first two terms so we'd have x times x plus 5 plus 6. Is this expression the same as this factor for it. Let's try and figure this thing out. I'm going to have you multiply these two factors and then we'll be able to determine if this is the same as this. If we distribute this x, x plus 6 and the that we get from the multiplication.
x times x would be x squared. X times 6 would be 6x. 5 times x would be positive
If we combine the like terms, we're left with x squared plus 11x plus 30. So, when we think about this factored form and this expression, is it equal to this original expression? Well, we know it's not equal. These are two different polynomials. This 6 wasn't multiplied by a factor of x plus 5 so we can't say that these two things are equal.
Let's look more closely at the product of these two factors and this polynomial. Do you remember that number puzzle that we've been playing? Well, there's a deep connection going on between these factors and this number puzzle. We know all three of these forms are equivalent. And for the number puzzle, I'm going to put a 30 in the top and 11 in the bottom. So what are the answers to this number puzzle? Write them in here.
I want to start by listing the factor pairs of 30. We have 1 and 30, 2 and 15, 3 and 10, and 5 and 6, and we could keep listing by putting 6 and 5. But we start to repeat a factor pair so we can stop. This lower number is positive 11. So, I know I don't need to have a negatives for both of the factors. I just want to find the factor pair that sums to 11. That factor pair, is 5 and 6. Good thinking if you found that factor pair.
So, let's make the connection between these equivalent forms and a Number Puzzle. What do you notice about the answers to this Number Puzzle? Which of these statements is true? Be sure to check all the statements that apply.
It turns out all of these are true. 5 and 6 sum to 11. 5 and 6 are also used as the coefficients of x to rewrite 11x. We know 5 and 6 is a factor pair of 30 since they multiply to give us 30. And finally we can see that a positive 5 and positive 6 up here in our binomials.
You might be wondering where the 30 and the 11 come from in the number puzzle. Well we actually multiply the coefficient of the x squared term and the constant term together to get this top number. This bottom number 11 comes from our x term. Whenever we want to factor a trinomial we can set up a number puzzle and then find the two factors we need to rewrite the x term. So, I can rewrite this expression as x squared plus 5x plus 6x plus 30. We can rewrite the 11x as 5x and 6x. The 2 numbers come from our number puzzle. By rewriting this x term as terms and a six from the second two terms. And whoa, look at that, our factored form. So for any quadratic trinomial, we can try this process to factor it. We can find two factors of a times c, that sum to the middle number b. We'll rewrite the middle term using those factors and then try factoring by grouping.
Try factoring this polynomial, using our techniques.
We start the number puzzle by multiplying 1 and 12. This gives us our top number. The x term has a coefficient of 7. So that goes inside the bottom. We need a factor pair of 12 that sums to 7. So that must be positive 3 and positive plus 7x plus 12 is equal to x squared plus 3x plus 4x plus 12. We can use factoring by grouping and remove an x from the first pair of terms and a 4 from the last pair of terms. This leaves us with x plus 4 times x plus 3
Here's a very similar polynomial, and I want you to try and factor it. Write your answer here.
We start by multiplying 1 and negative 12, the coefficient of the x squared term, and our constant term. That gives us the top number. Negative 4 is with the x term, so this goes in the bottom of our number puzzle. Here are the factor pairs for negative 12. I know one of the numbers needs to be negative in each of the pairs since a negative times a positive is a negative. We want to find the factor pair that sums to negative 4. That must be this pair, 2 and negative 6. Next we use our factor pair to rewrite the x term. We'll have x squared plus 2x plus negative 6x minus 12. I'm just showing you where these two factors come from in our number puzzle. But it's best to write this part as negative 6x. We can use factoring by grouping to remove an x from the first two terms, and a negative 6 from the last two terms. I need to remove a negative 6 from both these terms, so I wind up with x plus 2 for both of these parenthesis. Remember to distribute, to always have a quick check, to make sure you're doing the step right. We can factor one last time to get x minus 6 times x plus 2, our completely factored form.
As you factor more and more problems you want to look back at your work. Loot for patterns. If the product of the first coefficient and the constant term is positive, then both of the factors will have to be positive or they'll both have to be negative. If the product of the first coefficient and the last coefficient is negative, then we know one of these factors must be negative. We also see this in our factored form. Both of these are positive, where as here, one of them is negative.
Let's try factoring a harder trinomial. What number should we place in the top and bottom positions inside of our number puzzle in order to factor this trinomial?
Our polynomial is in the form ax squared plus bx plus c. We multiply the a and c together to get 12. This is the top number. This number will always come from the product of the coefficient of the x squared term and our constant. The b or the bottom number comes from our x term, the 13. Great work if you put 12 and
Finish factoring this polynomial by finding the factors needed to rewrite the x term. Once you've rewritten the x term, use factoring by grouping to finish up. And you might have noticed that this problem is slightly different because we have a 2 here. Our coefficient, our a value for the x squared term is not equal to one.
Here, the factor pairs for 12. And I know that each factor pair must be positive, since I have a positive 13 in the bottom. The only factor pair that adds to positive 13 is 1 and 12. This allows me to rewrite the middle term 13x as 1x plus 12x. Next we start factoring by grouping by taking out an x from the first two terms an a 6 from the second two terms. Finally, we factor out a 2x plus 1, leaving us with x plus 6 times 2x plus 1, our completely factored form. You're really improving your factoring skills if you got this correct.
What's so great about factoring is that we can check our solution. If I multiply these 2 binomials together, we would do this process in reverse and end up with our original problem. If you want to get faster at factoring trinomials, you want to find the factors of 2 times 6 or 12 that add up to 13. You always want to find factors of a times c that sum to b. Finding these 2 factors allows us to rewrite the middle term. And then allows us to use factoring by grouping.
Try factoring this polynomial. Write your answer here.
We start out number puzzle by multiplying 5 and 2, which equals 10. 7 is the x term, so that goes into the bottom. We want to find a factor pair of 10 that sums to 7. That must be 2 and 5. We can rewrite the 7x as positive 2x and positive 5x. Now we'll use factoring by grouping to finish out our problem. From the first two terms I can factor out an x, leaving me with 5x plus 2. The second two terms are a little bit tricky. They don't share a factor of 5, of x or 2, but they do share a factor of 1. We can always factor out a 1 if it appears that two terms do not have common factors. Our final factored form is x plus 1 times might have switched the 5x and the 2x and then continued factoring by grouping. That would work as well. As a quick check on the end, we can do some mental math to make sure this factored form is equal to this polynomial. X times 5x is 5x squared, 1 times 5x is 5x. And x times 2 is 2x. If I add 5x and 2x together, I'll get 7x. And finally, 1 times 2 is positive 2.
Whenever we want to factor a quadratic trinomial, we want to find two factors of a and c that add up to b. This allows us to rewrite our x term, like in 7x in the last problem, and then to continue using factoring by grouping. But our real first step is to see if there's a greatest common factor first. We should factor that out and then continue this process if we can.
Try factoring this polynomial. Write your answer in this box.
We want to find factors of a and c that add up to b, or 1 in this case. 2 times negative 15 is negative 30. And our positive 1 goes in the bottom. Here are the factor pairs for negative 30. Let the other side be the first factor negative in each pair. This factor pair adds up to 1, so I write negative 5 and 6. I rewrite the positive 1x using negative 5 and 6 as coefficients for x. We use factoring by grouping to remove an x and a positive 3 from the first group of terms and the second group of terms. This leaves us with our factored form x plus 3 times to the factor form.
For all the trinomials we've factored so far they had a Greatest Common Factor of 1. We haven't had to pull out a Greatest Common Factor first yet. Whenever we're given a trinomial we should first look to see if there is a Greatest Common Factor. What do you think is the Greatest Common Factor for this trinomial? Write that answer here. Then I want you to factor the GCF from this trinomial. You should have a GCF on the outside and then three terms in parenthesis here.
with this expression. Amazing algebra skills if you got that right. This will be the prime factorization for these three terms. We can see that 3x would be the greatest common factor. If we multiply what's remaining we'll get 12x squared for the first term, negative 5x for the second term, and negative 2 for the third term.
So, we know this trinomial can be factored like this. And now, we want to see if we can continue factoring this inside trinomial. So, if we wanted to continue factoring, which number should we use in our number puzzle? One of these numbers will be the top number, and then one of these will be the bottom number. Choose one from each row.
Since we've already factored the 3 x out, we want to work with this trinomial. This top number should be a times c. A is the coefficient in front of x squared and c is our constant term. So 12 times negative 2 equals negative 24, this choice. The bottom number is the b or the coefficient in front of x, so negative since I threw another polynomial at you. But, we want to make sure we work with factoring this quadratic.
So when trying to factor this polynomial, we want to rewrite the middle x term, by finding factors of negative 24, that sum to negative 5. Try finishing this number puzzle, and factoring this trinomial. The final answer will have all of the factors including the greatest common factor. So your answer should look like the GCF, a factor, and then another factor. Try your best and don't worry if it's not right on the first try.
Here are some factor pairs for negative 24. I made the first factor negative in each of the pairs. I know that none of these pairs could sum to negative 5 since the greater number is in the right hand column. All these sums would be positive. If I make the second factor negative, I can see that the sum of each of these factor pairs would end up being negative. So, the factor pair of negative 8 and 3 sum to negative 5. We can rewrite our negative 5x and negative and positive, it's generally easier to list the negative term first. This makes our factoring by grouping a little bit easier. We can factor our 4x from the first two terms, and a positive 1 from the second two terms, since the only factor that these two terms share is 1. I used brackets around this part of the factoring because I have parentheses inside. For this inside bracket portion, we have common factors of 3x minus 2, so we can factor that out, leaving us with our final complete factored form.
Let's see if we can put our knowledge together and try some practice problems. What do you think the factored form of this expression is?
We want to find factors of 10 that sum to negative 7. Well, the two factors must be negative 2 and negative 5. We can rewrite the middle term using minus 2x and minus 5x, from these factors. Then, we do our factoring by grouping and our final answer is x minus 5 times x minus 2. This stuff right here is the trickiest part. We need to factor out a negative 5 from both of these terms. If I factor out a negative 5 from positive 10, I'm left with negative 2. Remember that you can check your work by distributing. This allows us to be sure that we've done factoring by grouping correctly. Moving this way would be factoring, so if we distribute, we would be undoing that process. Nice work with your negative numbers and your factoring if you got that one correct.
Try factoring this expression.
This was the factored form. Excellent work if you got that one correct. When we start to factor, we should always ask ourself, is there GCF. Each of these terms doesn't share an a, but they do share a 5. If we factor out a 5, we're left with this trinomial. We multiply a times c to get negative 45. And then we put negative 4 in the bottom, the b term. We want to find factors of a times c that sum to b. So we want to find factors of negative 45 that sum to negative 4. Here are factor pairs for negative 45. I made the first factor negative, each of these sums would be positive so I need to make the second factors negative. 5 and negative 9 sum to negative 4. So this is what I use to rewrite negative 4a. Remember it's best to put the negative coefficient first. It makes our factoring more simple. We can factor 3a from the first two terms, and a positive 5 from the second two terms. Finally we factor out an a minus 3 to give us this expression. And we don't actually need to include these brackets anymore since all we have between factors is multiplication. This is the complete factored form.
Here's another practice problem. Good luck on this one.
For this trinomial, we want to find factors of a hundred, that sum to negative pairs for positive 100. But remember that this is positive 100, and I have a negative 20 down here. We need to find factors of positive 100 that sum to negative 20. This must mean that our factor pair must contain two negative numbers. So negative ten and negative ten must be the answers here. They multiply to give us positive 100 and they sum to negative 20. But before I go to do this step, I should have asked myself first, is there a GCF here? Well two doesn't go into every number because this one's not even. They don't all share an x, and there's no other common factors for 4, 20 and 25 besides one, so I can continue factoring as usual. We can factor 5x from the first two terms and a negative two from the second two terms. Completing my factoring by grouping, I have 5x minus 2 times 5x minus 2. Since it's factors repeated, I can simply write it with a square. Here's my complete factored form. Some of you might have taken a shortcut. You might have recognized this as a special pattern. Our first term and our last term are perfect squares. So this fits our perfect square trinomial pattern. We can just write out answer as 5x minus 2 squared. The a is
For this practice problem, try factoring this expression.
First we want to check to see if there's a greatest common factor other than 1, anything out. We want to find factors of negative 120 that sum to positive 14. Here's the start of a factor pair list. There are other factors for negative 120 but I already found the one I need, negative 6 and positive 20. We can check this real quick to make sure negative 6 times 20 multiplies to negative 120, and that they sum to positive 14. It's best to rewrite 14a with a negative term first. We finish out our problem by factoring by grouping, and these are two factors, 3a plus 4 and 5a minus 2. That's some great work if you got that one correct. If these are giving you trouble, stay with it. The more practice you do, the better you'll get at math. As you put in more effort, you will improve.
Here's another practice problem, try this one out.
First we look for a greatest common factor. The only factor shared between the three terms is 1, so we don't need to worry about that. We want to find the factor pair of 42 that sums to 17. That must be positive 3 and positive 14. We can rewrite 17x with 3x and 14x. Then we use factoring by grouping to get our final answer. Great work if you got all or even part of it correct. I know this factor pair was probably hard to find.
How about this one? What do you think the factor's would be for it? Write them here.
We want to start by looking for our Greatest Common Factor and what do you know all the coefficients are even, so 2 can come out. Now that we factored out a 2, we want to continue factoring this inside portion if possible. We want to find factors of positive 14 and some to negative 15. This 14 comes from multiplying the coefficient of the x squared term 2 by the constant term 7. The negative 15 is simply the coefficient of our x term. We have a positive 14 and a negative use negative 1 and negative 14 since they multiply to give us positive 14 and they add to negative 15. We continue factoring this portion by rewriting the middle term with these two factors as coefficients for x. Then we factor by grouping. We take out an x from the first two terms and a negative 7 from the second two terms. When I remove a negative 7 from positive 7, I'm left with negative 1 here. I also know I did this step correctly, since negative 7 times common factor of 2x minus 1. So this is our completed factor form. Fantastic work, if you got that correct.
Here's a challenge problem for you to solve. Now, there won't be a solution video posted, so feel free to discuss this with a friend or with other students on the forum. Write your factored form here, and good luck.
Here's another challenge problem that you can try. And again, this one won't have a solution video.
Here's the last challenge problem for this lesson.
I just wanted to say you guys are doing great so far. Congratulations on getting so far in visualizing algebra. Let's try another factoring trinomial problem. 2a cubed b, minus 28a squared b squared, plus 96ab cubed. Don't let the problem intimidate you. There's nothing to be afraid of. Just use what you've been learning to factor this trinomial. You can put your answer over here.
That was not an easy problem. Correct answer is 2ab times a minus 6b times a minus 8b. If you got it right, well done. If you didn't get it right, don't worry. We're going to go through it right now. Remember the first thing that you always want to do when factoring a trinomial is to look for the GCF. And that is the most common mistake that students make, is they forget to factor out the GCF. If you don't factor out the GCF first, you'll make your life a lot more difficult. Here, when I look at these 3 terms, first of all I noticed that all 3 of the numbers are even, and 2 is the largest number so 2 is going to part of our GCF. If I look at the variables I have an a cubed, an a squared, and an a, so the lowest power of a is to the first. I have a b, b squared and b cube to the lowest power of b is just b. When I factor out the GCF of 2ab from a polynomial, what I am left with? 2ab times a squared is meant 2a cube b. 2ab times minus 14ab, 2ab times negative 14ab, gives me negative 28 a squared, b squared. And 2 times 48 gives me 96. A, I need a b squared. So after I factor out the GCF of 2ab, I'm left with a squared minus 14ab plus 48b squared. Now I need to factor this polynomial. For this polynomial, since the coefficient of the a squared term is 1, I need the factors of 48 that add up to negative 14. Since this is a positive number and this is a negative number, I know that both of my factors are going to be negative. A negative times a negative is a positive and a negative plus a negative gives me a negative. When I think of the number 48, there are several ways to factor the number. The 2 factors of positive 48 that add up negative 14 is negative 6 times negative 8. So I can rewrite negative 14ab as negative 6ab minus 8ab. Now that I have 4 terms, I can factor by grouping. Don't forget to carry down your GCF. Now I need to look at the GCF of a squared and negative 6ab. Which is simply a. When I factor an a out, I'm left with a minus 6b. Now I need to factor out the GCF of the second two terms. I know I want my first term to be positive and my second to be negative. So I know I need to factor out a negative. The GCF of 8ab and 48b squared is going to be 8b. What do I need to multiply 8b by? What do I need to multiply negative 8b by to get negative 8ab? Just a. And what do I need to multiply negative 8b by to get positive 48b squared? Negative 6b. Now the last step for factoring by grouping here is to pull out the new GCF which is a minus know these problems aren't easy, but factoring is a key skill for the whole rest of the course. So it's very important that you're able to factor any polynomial to finish this course. So don't get discouraged, it takes a lot of practice. Keep up the good work. And don't worry, you will master factoring. Factoring is a key concept for the rest of this course. We will be using factoring to solve many additional problems throughout the rest of the course. So mastering factoring skill is extremely important. I know it's not easy, but the more you practice, the better you'll get. And you will master factoring.
I know you've already had some practice factoring some trinomials. Let's try a couple more problems. 50 x squared y squared minus 15 x y squared minus 20 y squared. Put your answer in this box.
The correct factorization of this trinomial is 5y squared times 5x minus 4 times you are on your way to being a factoring genius. Let's see how we got this answer. Remember, when I'm factoring any polynomial, I want to check to see if there's a greatest common factor. When I look at these three terms, I notice a couple of things. The first thing I notice is all three terms have a y-squared. In that case, I should be able to factor out a y-squared. The other thing that I notice is that each of the three terms is divisible by 5. So this polynomial has a GCF of 5 y-squared. When I factor out the 5 y-squared from each term, what am I left with? When I factor out the GCF, I'm left with 10 x squared, minus 3 x minus 4. 5 y squared times 10 x squared gives me the 50 x squared y squared. 5 y squared times negative 3 x is negative 15 x y squared, and negative 4 times 5 y squared gives me the negative 20 y squared. After I factor out the GCF, I need to look at the remaining polynomial, 10 x squared minus 3 x minus 4. Now I have to factor this trinomial. Remember when there is a coefficient in front of the x squared variable, you have to multiply the coefficient of the x squared variable by the constant term. To factor this polynomial, I have to look at the factors of 10 times negative 4 which is negative 40, that add up to negative 3. There are a lot of different factors that multiply together to get negative 40, but the factors that multiply together to get negative 40, and add together to get negative 3, are negative 8 and 5. Negative 8 times 5 is negative 40, negative 8 plus 5 is negative 3. Now we need to rewrite the middle term, using these two factors. When I rewrite negative 3x using my factors, I get 10x squared minus 8x plus 5x which gives me the negative 3x minus 4. Now I can factor by grouping. When I factor by grouping, I notice that the GCF of the first two terms is 2x, and I have 5x minus 4. 2x times 5x is 10x squared. 2x times negative 4 gives me negative 8x. Now this is an interesting situation. Here I have 5x minus 4, and that's what I want my second binomial to be. So what do I factor out? There is no GCF here. Well, technically there is a GCF, but the GCF is simply one. 1 times 5x, 5x. 1 times negative 4 is negative 4. Now I see the same binomial in both places, so I can now factor that out. Don't forget to bring along our overall GCF, 5x minus 4 times 2x plus 1. That was a pretty difficult problem. Great job if you got it right. If not maybe try it again.