Understanding Factoring Trinomials

In this video, we're going to talk about how to factor trinomials. In this example, we have a trinomial where the leading coefficient is 1. In order to factor it, you need to find two numbers that multiply to the constant term 12, but add to the middle coefficient, 7. So what are those two numbers? Well, let's begin by writing the factors of 12. If we divide 12 by 1, we get 12. If we divide 12 by 2, we get 6. If we divide 12 by 3, we get 4. Now which of these pairs of numbers multiply to 12 but add to 7? We know it's 3 and 4. 3 plus 4 is 7, but 3 times 4 is 12. So to factor the trinomial, we can write it as x plus 3 times x plus 4. And so that's how we could factor that particular trinomial. Now let's consider another example. So let's say we have x squared plus 11x plus 30. Go ahead and factor that particular trinomial. Feel free to pause the video if you want to. So first, we need to find two numbers that multiply to 30, but add to the middle coefficient, 11. So let's write down the factors of 30. If we divide 30 by 1, we get 30. If we divide it by 2, we get 15. 30 divided by 3 is 10. Now we can't divide 30 by 4 evenly. 4 doesn't go into 30. If you do 30 divided by 4, you're going to get 7.5. But we could divide it by 5 and get 6. And then if we continue further, if we divide it by 6, we get 5. Once the number is reversed, you can stop it there. Now which pair of numbers multiply to 30 but add to 11? But we know that 5 plus 6 add to 11. So the answer is going to be x plus 5 times x plus 6. So that's how we can factor x squared plus 11 x plus 30. That's what you can do if the leading coefficient is equal to 1. Now let's work on another example. This time we're going to solve a quadratic equation. So let's say we have the equation x squared minus 5x plus 6. equal to 0. How can we calculate the value of x that will make this equation true? Well since we have a trinomial on the left side we can factor it. So we need to find two numbers that multiply to 6 but add up to negative 5. If we divide 6 by 1 we get 6, if we divide it by 2 we get 3. Now 2 plus 3 add up to 5, but if we add two negative signs it will add to negative 5 and negative 2 times negative 3 is still equal. to positive 6. So to factor it's going to be x minus 2 times x minus 3. So now that we've factored the trinomial we can now solve the equation. Using the zero product property we can set each factor equal to zero. For the first one, if we add 2 to both sides, we'll get that x is equal to 2. For the second one, if we add 3 to both sides, we'll get that x is equal to 3. And so these are the two solutions to this quadratic equation. Try this example. Let's say that x squared plus 3x. minus 28 is equal to 0. Go ahead and solve for this particular quadratic equation. So let's begin by factoring it. Let's find two numbers that multiply to negative 28, but add to positive 3. So if we divide 28 by 1, we're going to get negative 28. If we divide it by 2, we'll get negative 14. 3 doesn't go into 28, but we can divide it by 4 and get negative 7. And then if we divide it by 7, we'll get negative 4. And we'll stop there. Now notice that 4 plus negative 7 adds up to negative 3. However, positive 7 plus negative 4 adds up to positive 3. And this is what we want to use. So when factoring it, it's going to be x plus 7 times x minus 4. Now let's set each factor equal to 0. Let's go ahead and solve this equation. So first, let's subtract both sides by 7. And this will give us the first answer, x is equal to negative 7. And for the second one, if we add 4 to both sides, we get x is equal to 4. So we have our two possible solutions, x is equal to negative 7 or x is equal to 4. And what you could do to check your work is you can plug it back in to the original equation. Let's plug in 4. 4 squared plus 3 times 4 minus 28. Let's see if that's going to give us 0. 4 squared or 4 times 4 is 16. 3 times 4 is 12. 16 plus 12 is 28. 28 minus 28 is 0. So we can see that this indeed works. Now let's work on an example where the leading coefficient is not equal to 1. In this example, the leading coefficient is 2. How can we solve this particular quadratic equation? How can we factor the trinomial? Well, to do that, what you need to do is multiply the leading coefficient by the constant term. 2 times 6 is equal to 12. Next, you need to find two numbers that multiply to 12 but add to the middle coefficient, negative 7. So let's write down the factors of 12. It's 1 and 12, 2 and 6, 3 and 4. Now, 3 plus 4 add up to positive 7, but if we use negative 3 and negative 4, it will add up to negative 7 and still multiply to positive 12. Now, for this type of problem, we can't just write the answer immediately. There's going to be some additional steps here. What we're going to do now that's different is we're going to replace the middle term with these two. So you can write negative 3x minus 4x, or you can write negative 4x minus 3x. I'm going to write it this way, negative 4x minus 3x. The reason being is these two are even numbers, and you can take out a 2, and for those two numbers, they're multiples of 3. You can easily factor out a 3. Now, let's factor by grouping. In the first two terms, we're going to take out the GCF, which is 2x. If you divide 2x squared by 2x, you're going to get x. If you divide negative 4x by 2x, you're going to get negative 4x. 2. Now in the next two terms we're going to take out the GCF or the greatest common factor which is going to be negative 3. Negative 3x divided by negative 3 is x. Positive 6 divided by negative 3 is negative 2. Now, if these two are the same, you know that you're on the right track. So we're going to factor the GCF out of these two terms. And the GCF is what I have highlighted in blue. It's x minus 2. So if you take out x minus 2 from that term, you're left with 2x. If you take out x minus 2 from this term, you're left with minus 3. So this is how you can factor a trinomial where the leading coefficient is not 1. Now to check our work, what we could do is FOIL. If we FOIL these two binomials, we should end up with a trinomial that we started with. A binomial is basically a polynomial expression with two terms. A trinomial is one with three terms. 1, 2, 3. I had four marks, but this is 1, 2, 3. If we multiply x by 2x, we're going to get 2x squared. If we multiply x by negative 3, we're going to get negative 3x. Negative 2 times 2x is negative 4x. Negative 2 times negative 3 is plus 6. And then these two here will add up to negative 7x. As you can see, these two expressions are equivalent. So that tells us that we are on the right track. Now the last thing we need to do is solve the equation. So let's set each factor equal to 0 using the 0 product property. So if we add 2 to both sides for the first equation, we're going to get the first solution, which is x is equal to 2. For the second one, if we add 3 to both sides, We'll get that 2x is equal to 3, and then we'll have to divide both sides by 2. So we get the solution, x is equal to 3 over 2. So here are the two answers, x is equal to 2, and x is equal to 3 over 2. Now, for the sake of practice, let's work on another example. And for this one, we're not going to try to solve the equation, because you already know how to do that at this point. We're simply going to focus on factoring the trinomial. Well, since the leading coefficient is not 1, it's 3 this time, we're going to multiply the leading coefficient by the constant term. So 3 times negative 6 is negative 18. And then we're going to look for two numbers that multiply to negative 18, but add to the coefficient, the middle coefficient, of negative 7. So let's write down the factors of negative 18. If we divide it by 2, we're going to get negative 9. If we divide it by 3, we'll get negative 6. And if we divide it by 6, we'll get negative 3. Now note that 2 and negative 9, they multiply to negative 18, and they add up to the middle coefficient, negative 7, which is what we want. So we can replace negative 7x with positive 2x and negative 9x, or we can reverse it and write it as negative 9x plus 2x. In either case, these two, they still add up to negative 7x. But I chose to put it that way because when factored by grouping, it's just going to be a lot easier, in my opinion. Now let's take out the GCF in the first two terms. So that's going to be 3x. 3x squared divided by 3x is x. Negative 9x divided by 3x is negative 3. Now for the last two terms, let's take out the GCF, which is going to be 2. 2x divided by 2, that's x. Negative 6 divided by 2 is negative 3. Next, let's factor out x minus 3. So if we take out x minus 3 in the first term, we're going to have 3x. And then if we take it away in the second term, we're going to have 2. So this is the answer. x minus 3 times 3x plus 2. And just to check it, x times 3x is 3x squared, x times 2 is 2x, negative 3 times 3x is negative 9x, negative 3 times 2 is negative 6. And then 2x minus 9x is negative 7x. So we start with the original expression. So this right here is the answer. Now let's try a challenging problem. So consider the trinomial 72x squared plus 17x minus 70. How would you factor this particular trinomial? Well, if you follow the steps that we've been using, you know the first thing to do is to multiply the leading coefficient by the constant term. 72 times negative 70, that's negative 5040. Now, it can be challenging to find two numbers that multiply to 5040, or negative 5040, and at the same time, add up to 17. Now, the best way to do this is to do it systematically and find a, just go by the list. Let's say if you divide this by 1, you get negative 5040. If you divide it by 2, you get negative 2040. 520. And then if you keep doing this, eventually you're going to find two numbers that will multiply to negative 5040 and add to 17. But that's going to take a lot of time. So is there a more efficient way in which we can factor this trinomial? It turns out that there is. And it involves using the quadratic formula. Let's set this equal to zero. Let's begin by writing the quadratic formula. It's equal to negative b plus or minus the square root of b squared minus 4ac divided by 2a. a is the leading coefficient, 72. b is the number in front of x. c is the constant term. So in this example, we have negative 17 plus or minus the square root of 17 squared minus 4 times a times c divided by 2a or 2 times 72. So let's simplify it. 17 squared is 289. And if we multiply negative 4 by 72 times negative 70, that's going to be 20,160. And 2 times 72 is 144. Now, 289 plus 20,160, that's 20,449. If you take the square root of that number, you're going to get 143. Now we can break this down into two answers. The first one is going to be, keep in mind we have two signs. So the first one is going to be negative 17 plus 143 divided by 144. Negative 17 plus 143, that's 126. Now we need to reduce this fraction. We could divide both numbers by 2. 126 divided by 2 is 63. 144 divided by 2 is 72. Now 63 is 9 times 7. 72 is 9 times 8. So this reduces to 7 over 8. So this is one of the answers to the equation. x is 7 over 8. Now let's get the other answer. So instead of using the plus sign, we're now going to use the minus sign. Negative 17 minus 143 over 144. Negative 17 minus 143. That is negative 160. Now let's see how we can break this into smaller numbers. If we divide by 2, half of negative 160 is negative 80, half of 144 is 72. So these are numbers that we can deal with. So we know that 72 is 9 times 8. And 80 is 10 times 8. And let's not forget the negative sign. So if we cancel an 8, we're left with negative 10 over 9. So that's the second solution to this equation. Now remember, our goal is not necessarily to solve the quadratic equation, because we have those answers. But our goal is to find an easy way to factor this expression. And basically, we're reverse factoring it using the quadratic formula. So here's what we're going to do now. We're going to write each solution. Now from this point, we need to get a 0 on one side of the equation. To do that, I'm going to multiply both sides by 8 for the first equation. Here the 8s will cancel and I get 8x is equal to 7. Next, I'm going to subtract both sides by 7. So on the left, I'm going to get 8x minus 7, and on the right, 0. You can't really combine these terms because they are unlike terms. Now let's do the same thing for the other solution. Let's multiply both sides by 9. This will give us 9x is equal to negative 10. Now I'm going to move the negative 10 from the right side to the left side. As I move it from the right side to the left side, it's going to change side. It's going to go from negative 10 to positive 10. And so I'm going to get 9x plus 10 is equal to 0. So these two things right here are the factors to the trinomial. So thus to factor the trinomial, we can simply write 8x minus 7 times 9x plus 10. And so that's how we can reverse factor the trinomial using the quadratic formula. So this right here is the answer. And if we want to check it, just to make sure we have it right, it's going to be 8x times 9x, which is 72x squared, and then 8x times 10, that's 80. X negative 7 times 9 X that's negative 63 X negative 7 times 10 is negative 70 and then if we combine the two middle terms 80 minus 63 is 17 And so we get the original expression that we started with. So now you know how to reverse factor a trinomial using the quadratic formula.

We know it's 3 and 4. 3 plus 4 is 7, but 3 times 4 is 12. So to factor the trinomial, we can write it as x plus 3 times x plus 4. And so that's how we could factor that particular trinomial. Now let's consider another example. So let's say we have x squared plus 11x plus 30. Go ahead and factor that particular trinomial.

Feel free to pause the video if you want to. So first, we need to find two numbers that multiply to 30, but add to the middle coefficient, 11. So let's write down the factors of 30. If we divide 30 by 1, we get 30. If we divide it by 2, we get 15. 30 divided by 3 is 10. Now we can't divide 30 by 4 evenly. 4 doesn't go into 30. If you do 30 divided by 4, you're going to get 7.5. But we could divide it by 5 and get 6. And then if we continue further, if we divide it by 6, we get 5. Once the number is reversed, you can stop it there.

Now which pair of numbers multiply to 30 but add to 11? But we know that 5 plus 6 add to 11. So the answer is going to be x plus 5 times x plus 6. So that's how we can factor x squared plus 11 x plus 30. That's what you can do if the leading coefficient is equal to 1. Now let's work on another example. This time we're going to solve a quadratic equation.

So let's say we have the equation x squared minus 5x plus 6. equal to 0. How can we calculate the value of x that will make this equation true? Well since we have a trinomial on the left side we can factor it. So we need to find two numbers that multiply to 6 but add up to negative 5. If we divide 6 by 1 we get 6, if we divide it by 2 we get 3. Now 2 plus 3 add up to 5, but if we add two negative signs it will add to negative 5 and negative 2 times negative 3 is still equal.

to positive 6. So to factor it's going to be x minus 2 times x minus 3. So now that we've factored the trinomial we can now solve the equation. Using the zero product property we can set each factor equal to zero. For the first one, if we add 2 to both sides, we'll get that x is equal to 2. For the second one, if we add 3 to both sides, we'll get that x is equal to 3. And so these are the two solutions to this quadratic equation. Try this example.

Let's say that x squared plus 3x. minus 28 is equal to 0. Go ahead and solve for this particular quadratic equation. So let's begin by factoring it.

Let's find two numbers that multiply to negative 28, but add to positive 3. So if we divide 28 by 1, we're going to get negative 28. If we divide it by 2, we'll get negative 14. 3 doesn't go into 28, but we can divide it by 4 and get negative 7. And then if we divide it by 7, we'll get negative 4. And we'll stop there. Now notice that 4 plus negative 7 adds up to negative 3. However, positive 7 plus negative 4 adds up to positive 3. And this is what we want to use. So when factoring it, it's going to be x plus 7 times x minus 4. Now let's set each factor equal to 0. Let's go ahead and solve this equation. So first, let's subtract both sides by 7. And this will give us the first answer, x is equal to negative 7. And for the second one, if we add 4 to both sides, we get x is equal to 4. So we have our two possible solutions, x is equal to negative 7 or x is equal to 4. And what you could do to check your work is you can plug it back in to the original equation.

Let's plug in 4. 4 squared plus 3 times 4 minus 28. Let's see if that's going to give us 0. 4 squared or 4 times 4 is 16. 3 times 4 is 12. 16 plus 12 is 28. 28 minus 28 is 0. So we can see that this indeed works. Now let's work on an example where the leading coefficient is not equal to 1. In this example, the leading coefficient is 2. How can we solve this particular quadratic equation? How can we factor the trinomial? Well, to do that, what you need to do is multiply the leading coefficient by the constant term. 2 times 6 is equal to 12. Next, you need to find two numbers that multiply to 12 but add to the middle coefficient, negative 7. So let's write down the factors of 12. It's 1 and 12, 2 and 6, 3 and 4. Now, 3 plus 4 add up to positive 7, but if we use negative 3 and negative 4, it will add up to negative 7 and still multiply to positive 12. Now, for this type of problem, we can't just write the answer immediately.

There's going to be some additional steps here. What we're going to do now that's different is we're going to replace the middle term with these two. So you can write negative 3x minus 4x, or you can write negative 4x minus 3x.

I'm going to write it this way, negative 4x minus 3x. The reason being is these two are even numbers, and you can take out a 2, and for those two numbers, they're multiples of 3. You can easily factor out a 3. Now, let's factor by grouping. In the first two terms, we're going to take out the GCF, which is 2x. If you divide 2x squared by 2x, you're going to get x.

If you divide negative 4x by 2x, you're going to get negative 4x. 2. Now in the next two terms we're going to take out the GCF or the greatest common factor which is going to be negative 3. Negative 3x divided by negative 3 is x. Positive 6 divided by negative 3 is negative 2. Now, if these two are the same, you know that you're on the right track.

So we're going to factor the GCF out of these two terms. And the GCF is what I have highlighted in blue. It's x minus 2. So if you take out x minus 2 from that term, you're left with 2x.

If you take out x minus 2 from this term, you're left with minus 3. So this is how you can factor a trinomial where the leading coefficient is not 1. Now to check our work, what we could do is FOIL. If we FOIL these two binomials, we should end up with a trinomial that we started with. A binomial is basically a polynomial expression with two terms.

A trinomial is one with three terms. 1, 2, 3. I had four marks, but this is 1, 2, 3. If we multiply x by 2x, we're going to get 2x squared. If we multiply x by negative 3, we're going to get negative 3x. Negative 2 times 2x is negative 4x. Negative 2 times negative 3 is plus 6. And then these two here will add up to negative 7x.

As you can see, these two expressions are equivalent. So that tells us that we are on the right track. Now the last thing we need to do is solve the equation.

So let's set each factor equal to 0 using the 0 product property. So if we add 2 to both sides for the first equation, we're going to get the first solution, which is x is equal to 2. For the second one, if we add 3 to both sides, We'll get that 2x is equal to 3, and then we'll have to divide both sides by 2. So we get the solution, x is equal to 3 over 2. So here are the two answers, x is equal to 2, and x is equal to 3 over 2. Now, for the sake of practice, let's work on another example. And for this one, we're not going to try to solve the equation, because you already know how to do that at this point. We're simply going to focus on factoring the trinomial. Well, since the leading coefficient is not 1, it's 3 this time, we're going to multiply the leading coefficient by the constant term.

So 3 times negative 6 is negative 18. And then we're going to look for two numbers that multiply to negative 18, but add to the coefficient, the middle coefficient, of negative 7. So let's write down the factors of negative 18. If we divide it by 2, we're going to get negative 9. If we divide it by 3, we'll get negative 6. And if we divide it by 6, we'll get negative 3. Now note that 2 and negative 9, they multiply to negative 18, and they add up to the middle coefficient, negative 7, which is what we want. So we can replace negative 7x with positive 2x and negative 9x, or we can reverse it and write it as negative 9x plus 2x. In either case, these two, they still add up to negative 7x.

But I chose to put it that way because when factored by grouping, it's just going to be a lot easier, in my opinion. Now let's take out the GCF in the first two terms. So that's going to be 3x.

3x squared divided by 3x is x. Negative 9x divided by 3x is negative 3. Now for the last two terms, let's take out the GCF, which is going to be 2. 2x divided by 2, that's x. Negative 6 divided by 2 is negative 3. Next, let's factor out x minus 3. So if we take out x minus 3 in the first term, we're going to have 3x.

And then if we take it away in the second term, we're going to have 2. So this is the answer. x minus 3 times 3x plus 2. And just to check it, x times 3x is 3x squared, x times 2 is 2x, negative 3 times 3x is negative 9x, negative 3 times 2 is negative 6. And then 2x minus 9x is negative 7x. So we start with the original expression.

So this right here is the answer. Now let's try a challenging problem. So consider the trinomial 72x squared plus 17x minus 70. How would you factor this particular trinomial?

Well, if you follow the steps that we've been using, you know the first thing to do is to multiply the leading coefficient by the constant term. 72 times negative 70, that's negative 5040. Now, it can be challenging to find two numbers that multiply to 5040, or negative 5040, and at the same time, add up to 17. Now, the best way to do this is to do it systematically and find a, just go by the list. Let's say if you divide this by 1, you get negative 5040. If you divide it by 2, you get negative 2040. 520. And then if you keep doing this, eventually you're going to find two numbers that will multiply to negative 5040 and add to 17. But that's going to take a lot of time. So is there a more efficient way in which we can factor this trinomial? It turns out that there is.

And it involves using the quadratic formula. Let's set this equal to zero. Let's begin by writing the quadratic formula.

It's equal to negative b plus or minus the square root of b squared minus 4ac divided by 2a. a is the leading coefficient, 72. b is the number in front of x. c is the constant term. So in this example, we have negative 17 plus or minus the square root of 17 squared minus 4 times a times c divided by 2a or 2 times 72. So let's simplify it.

17 squared is 289. And if we multiply negative 4 by 72 times negative 70, that's going to be 20,160. And 2 times 72 is 144. Now, 289 plus 20,160, that's 20,449. If you take the square root of that number, you're going to get 143. Now we can break this down into two answers. The first one is going to be, keep in mind we have two signs.

So the first one is going to be negative 17 plus 143 divided by 144. Negative 17 plus 143, that's 126. Now we need to reduce this fraction. We could divide both numbers by 2. 126 divided by 2 is 63. 144 divided by 2 is 72. Now 63 is 9 times 7. 72 is 9 times 8. So this reduces to 7 over 8. So this is one of the answers to the equation. x is 7 over 8. Now let's get the other answer. So instead of using the plus sign, we're now going to use the minus sign. Negative 17 minus 143 over 144. Negative 17 minus 143. That is negative 160. Now let's see how we can break this into smaller numbers.

If we divide by 2, half of negative 160 is negative 80, half of 144 is 72. So these are numbers that we can deal with. So we know that 72 is 9 times 8. And 80 is 10 times 8. And let's not forget the negative sign. So if we cancel an 8, we're left with negative 10 over 9. So that's the second solution to this equation. Now remember, our goal is not necessarily to solve the quadratic equation, because we have those answers. But our goal is to find an easy way to factor this expression.

And basically, we're reverse factoring it using the quadratic formula. So here's what we're going to do now. We're going to write each solution.

Now from this point, we need to get a 0 on one side of the equation. To do that, I'm going to multiply both sides by 8 for the first equation. Here the 8s will cancel and I get 8x is equal to 7. Next, I'm going to subtract both sides by 7. So on the left, I'm going to get 8x minus 7, and on the right, 0. You can't really combine these terms because they are unlike terms. Now let's do the same thing for the other solution. Let's multiply both sides by 9. This will give us 9x is equal to negative 10. Now I'm going to move the negative 10 from the right side to the left side.

As I move it from the right side to the left side, it's going to change side. It's going to go from negative 10 to positive 10. And so I'm going to get 9x plus 10 is equal to 0. So these two things right here are the factors to the trinomial. So thus to factor the trinomial, we can simply write 8x minus 7 times 9x plus 10. And so that's how we can reverse factor the trinomial using the quadratic formula.

So this right here is the answer. And if we want to check it, just to make sure we have it right, it's going to be 8x times 9x, which is 72x squared, and then 8x times 10, that's 80. X negative 7 times 9 X that's negative 63 X negative 7 times 10 is negative 70 and then if we combine the two middle terms 80 minus 63 is 17 And so we get the original expression that we started with. So now you know how to reverse factor a trinomial using the quadratic formula.

Transcript for:Understanding Factoring Trinomials

Transcript for:
Understanding Factoring Trinomials