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In this video, we're going to learn about factoring trinomials of the form
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x squared plus b x plus c. Let's look at this expression, x minus two times x plus five. If
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we multiply these two binomials together, which we have already learned how to do,
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we will get the trinomial x squared plus three x minus ten. This form where we have
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the two binomials multiplied together is known as factored form. The factored form of a quadratic
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expression, as this is, is the product of two linear factors and possibly a constant.
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Let's see if we can determine some relationships between this factored form and this trinomial.
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As I look at this factored form, I notice that the first terms of my
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two binomials multiply together to give me the first term in my trinomial, x squared.
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I also notice that these two pieces of my binomials minus two and five are
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factors of this last term in my trinomial. They are factors of negative ten.
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Further, if I add these two numbers, negative two and five, that is the coefficient that is in my
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middle term or my b x term. We can use these patterns when helping us determine whether or
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not the trinomial can be factored. We'll look for two numbers whose product is c and whose sum is b.
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If a quadratic cannot be factored over the integers, we say that it is prime. Let's
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look at some examples. We want to factor each trinomial or state that the expression is prime.
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Our first example is x squared plus five x plus six. We notice that this is in the form
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x squared plus b x plus c, and there are no common factors in these terms. We always want to look
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for a greatest common factor when factoring first, prior to moving into this next step of factoring.
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Now to determine how this factors, I'm going to set up two linear factors, and I notice that the
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first term in this trinomial is x squared. So my first term of each of these binomials
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will be x. When I multiply x times x, we get x squared. To determine the second term of each of
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these binomials, I'm going to first look at this constant, six, and I'm looking for two numbers who
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are factors of six, and those two numbers must sum to give me five, which is the b value here.
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So let's think of all of the different ways we can factor six. I know that one times six gives me
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six. But if I add one and six together, I do not get a sum of five. I know that two times three
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also gives me six, and when I add two and three, then I do get the correct sum of positive five. So
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we will use these two numbers, a positive two and a positive three, to finish our binomial factors.
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Now the order of these two linear factors does not matter because multiplication is commutative.
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So the factored form of x squared plus five x plus six is x plus two
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times x plus three. You can always check that you have factored correctly by using the
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distributive property to multiply this out, and you will get the trinomial that you began with.
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Let's try another one. In this problem, we have m squared minus seventeen m plus seventy. So I'm
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going to set up my two linear factors. Again, I notice that there is no greatest common factor
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or no common factor of these three terms. My first factor is, my first term rather is,
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m squared. And so the first terms of my binomial will be m and m. m times m gives me m squared.
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Now to find the second terms of these binomials, I'm looking for two numbers or two factors
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that give me the product of seventy, and they must add together to give me the sum
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of negative seventeen. Think about what those factors might be. You might try
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one and seventy and realize that that is way too big of a sum. That's seventy-one.
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Or two and thirty-five, again you're much too high when you add those numbers together.
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Continuing to think through the factors of seventy, you might get to seven and ten. What happens when you add
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seven and ten? Well, we get seventeen, a positive seventeen, but what I'm looking for is a negative seventeen.
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If I change the sign here of each of those factors, negative seven and negative ten,
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I see that the product is still positive seventy and the sum is negative seventeen,
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just as we want it. So the second term of each of our binomials will be these two
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values, minus seven and minus ten. So the factored form of m squared minus seventeen m plus seventy
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is m minus seven times m minus ten. Let's continue our work with factoring. Next, we want to factor a squared
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minus two a minus forty-eight. We recognize that this is in the form x squared plus b x plus c, and there are
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no common factors in these three terms. So we'll set up our two linear factors.
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And as we consider the first term of these linear factors, we look to the first term of our
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trinomial which is a squared. That means that the first term of each of our binomials will be a. a
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times a gives me a squared. As we consider what might be the second terms of our binomials, we
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need to look at the value of c which is negative forty-eight. So I need two numbers that multiply
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together to give me negative forty-eight, and they must add together to give me negative two. Now
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we're looking for a product that is negative. This means that one of the factors must be positive
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while the other factor is negative because a positive times a negative gives us a negative.
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So we may start to think of all the ways we can make forty-eight. One and forty-eight,
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with one of them being negative, we would choose the larger number to be negative so that the sum
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is also negative. But notice that this sum is much too high. This gives us negative forty-seven. Two
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and twenty-four, with the twenty-four being negative, still the sum would be much too high.
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Three and sixteen, four and twelve, six and eight. And if the eight were negative, then this sum gives us negative two.
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So these are the two numbers that finish the binomials in this factored form.
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We have a positive six and a negative eight. Again, the order of the two binomials does not matter, so
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I could have written this as a minus eight times a plus six. And if I multiply these out using the
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distributive property, I will see that I get this trinomial a squared minus two a minus forty-eight.
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In our next example, we have x squared plus two x minus ten. As we begin to try to factor this,
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we would think that we would have two linear factors and that the first term
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of those linear factors would be x in order that I get x squared for my first term.
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And then I look to my constant term negative ten. And I'm trying to find two numbers that
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multiply together to give me negative ten, and they must add together to give me a positive two.
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Let's think of the ways that I can make negative ten by multiplying two numbers together. I can do
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one and ten. In this case, if I'm trying to get a positive sum and I want a negative product,
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then the larger number would be positive. So I would have a negative one and a positive ten,
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and that sum would be positive nine. The only other way that we can make ten
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using the product of two numbers is two and five. And again, to get a positive sum, that two
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would be negative. But this sum is not positive two. This sum is positive three.
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There are no other factors of negative ten. There are no factors that multiply together
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to give me negative ten and add together to give me positive two. Thus, this trinomial
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can't be factored over the integers. When that is the case, we say that the trinomial is prime.
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Let's take a look at another example. Here we have x squared plus x y minus twelve y squared. This trinomial is in the form
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x squared plus b x plus c, in this case though my c is a little different looking.
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It's negative twelve y squared is my c, and my b value which is the factor in front of the x
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is y. We'll begin by setting up the factors of our two binomials the same way we have for the others.
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I notice that my first term is x squared in the trinomial, so my first two terms of my binomials
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will be x because x times x is x squared. Now in finding the second terms of these binomials,
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I need two numbers that multiply together to give me negative twelve y squared,
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which is our c value, and they need to add together to give me y or one y. What would I multiply these two numbers together?
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If I'm going to get a sum of y, I would like for each of these factors to include y,
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so that I can add them together as like terms. So let's really focus on this
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negative twelve. What ways can I make negative twelve by multiplying two numbers together?
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One and twelve, six and two, three and four. I notice that I have to have different signs to get a
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negative product, and I need a positive sum. So the larger of the two numbers will be positive.
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So we can use four y and negative three y. The sum of four y and negative three y
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is a positive one y. So this will give me x plus four y times x minus three y.
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And in our last example here, we have sixteen b plus fifteen plus b squared.
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Is this in the form x squared plus b x plus c? Well, it's got the right terms,
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but they're not in the right order. So let's begin by rearranging this trinomial.
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This can be written as b squared plus sixteen b plus fifteen. Remember that addition is commutative, so I
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can rearrange the order of these terms. And now I can recognize that this is in the form x squared plus b x plus c,
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where b is sixteen and c is fifteen. So we're going to need two numbers
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that multiply together to give me fifteen and their sum is sixteen.
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This would be fifteen and one. As I write the factored form with its linear factors, I begin by looking at
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the b squared and recognizing that each of these binomials must contain a b to the first power,
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and then I use my factors fifteen and one to complete the binomials, leaving me with
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b plus fifteen times b plus one as the factored form of this trinomial.
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Recall from previous videos and sections that we begin our factoring by looking for a GCF or
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greatest common factor as we factor our expressions. So when we're looking
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at these examples, the directions are to factor each expression completely.
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And the first example is five t squared minus sixty t plus one hundred sixty. Looking at those three terms, you should
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notice a common factor of five. The GCF of this expression is five. So we'll begin by factoring out a five. And to
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determine what's left in these grouping symbols, we simply divide each of these three terms by five.
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This will leave us with t squared minus twelve t plus thirty-two. So we have factored out the GCF,
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but we have not yet factored completely. Notice that what is left here is a trinomial of the form
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x squared plus b x plus c, and it is possible that this trinomial can be factored. So let's see.
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If it can be factored, it's a quadratic, so that means we would have two linear factors.
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And the first term of these binomials would be t because t times t gives me t squared. To find
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the second term of each of these binomials, I look to my constant term thirty-two, and
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I'm looking for factors of thirty-two that sum together to give me negative twelve.
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So if I'm looking for two numbers that multiply together to give me a positive number and they
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add together to give me a negative number, then those two numbers both need to be negative. Let's
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think of the different ways that we can make thirty-two by multiplying two numbers together.
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We could do one times thirty-two, or negative one times negative thirty-two, and that sum would be much too large.
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We could do two and sixteen. Again, the sum is too large. We could do four and eight.
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Four times eight does give me thirty-two, but the sum is positive twelve. And as noted before,
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in order to get a negative sum here, I'm going to need to have negative four and negative
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eight. That gives me a positive thirty-two as a product and a negative sum of twelve.
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So those will be the second terms of each of these binomials. We'll have t minus four times
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t minus eight. But let's not forget that we factored out a GCF in the first step.
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And we can't leave that out. So we must put that back. And our final factored form is five times t
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minus four times t minus eight. We could check this by using the distributive property to
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multiply, and we would get this original trinomial. Let's look at our next example. We have three y
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cubed minus eighteen y squared minus twenty-one y. Let's look for the
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GCF of these three terms. Well, I notice that three, eighteen, and twenty-one have
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a common factor of three, and y cubed, y squared, and y have a y in common. So the common factor,
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or the greatest common factor, of all three terms is three y. And what's remaining when I factor a three y from
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these three terms is y squared minus six y minus seven. So this is a factored form, but it is not completely factored
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because I see that I have a trinomial here in these grouping symbols that is a quadratic that may be factored further.
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And so if it can be factored further, it will factor into two linear factors. The first
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term of each will be y because y times y is y squared. And to come up with the second terms,
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we need two numbers that multiply together to give me negative seven, and they must add together
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to give me negative six. How do we make a product of a negative seven with two numbers?
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One of those numbers has to be negative, and one must be positive. So I know that one and
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seven are factors. Which one needs to be negative in order to get a negative sum?
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One plus negative seven gives us negative six. So these are our factors. We will have y
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plus one and y minus seven as our linear factors. And we must not forget the GCF of three y. So our
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completely factored form here of this expression is three y times y plus one times y minus seven.
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And here's our last expression, our last example. Negative x squared plus nine x
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minus fourteen. The only common factor in this trinomial is one or negative one.
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In general, when your trinomial or expression begins with a negative number,
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it's easiest to factor if you first factor out a negative one. So we're going to begin there.
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We will factor out a negative one, and then this will give us x squared minus nine x
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plus fourteen as the remaining trinomial inside those grouping symbols.
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It's a quadratic, and so we can factor that quadratic with two linear factors. The first term
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of each will be x. And for the last term I need two numbers that multiply together to give me
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positive fourteen, and they must add together to give me negative nine. What two numbers multiply
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together to give me positive fourteen and add together to give me negative nine?
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How about negative two and negative seven? So our completely factored form will be negative one times x minus two times x minus seven.