Integrated Math 2–3 Honors — Factoring Polynomials with GCF and Grouping
Corrected lesson transcript


## 0:00 — Factoring Polynomials with GCF and Grouping


## 0:05 — Multiplication and Its Reverse

[0:05] What are the possible directions for this problem? It could say simplify or multiply.
[0:21] Just multiply, which we need to use the distributive property.
[0:31] And of course we now need the distributive property of multiplication over the two additions.
[0:37] Okay, and there are no like terms.
[0:41] That would be it.
[0:43] You could be writing descending order.
[0:48] Okay, so today we're doing the opposite, or the reverse.
[0:53] First, we're going to start with this and we want to end with this.
[1:06] What's the reverse process called?
[1:09] It's called factoring.

## 1:10 — Factor vs. To Factor

[1:10] I'll start with the definition. There's actually two definitions of the word
[1:14] To factor means to write an expression as a product.
[1:35] Right, this is not a product, this is a sum.
[1:42] This polynomial, first off, what would this polynomial be called?
[1:47] A fourth degree trinomial.
[1:49] know if we're able to write it as a
[1:53] And we give you the number 24.
[1:56] And I said factor.
[1:58] What does factor mean?
[2:03] To write a product.
[2:05] So what are some possible answers here?
[2:11] 1 times 24.
[2:15] 2 times 12.
[2:18] 3 times 8.
[2:23] 4 times 6.
[2:27] 6 times 4.
[2:29] Same sort of thing.
[2:31] Is that it?
[2:35] How about 2 times 2 times 6?
[2:43] How about 2 times 2 times 2 times 3?
[2:54] How about...
[2:55] What else do I got?
[2:57] 2 times 3 times 4?
[3:02] Is that right?
[3:07] There's a lot of ways to write this as a product. These are often called factor pairs.
[3:17] Is there two numbers? Two different factors? These are often not. So, factor is kind of ambiguous.
[3:27] okay what it means to factor because there's lots of different possibilities okay

## 3:36 — Prime Factorization

[3:36] This is called the prime factorization: 2 cubed times 3 squared.
[4:03] factored it into its prime factorization. The product of all its prime factors lead
[4:12] to the correct exponent. Okay. Why don't I put a number 1 in front? True. Well, what
[4:32] if it's one squared. Is that fine? No? Is it the same thing? What about one cube?
[4:42] They're all equal. But the reason we don't put a 1, because the number 1 is not
[4:49] prime, it's not a prime number. Why is the number 1 not a prime number?
[5:05] What is the definition of a prime number?
[5:09] Numbers prime yet?
[5:16] It's only divisible by one in itself?
[5:21] Or only be written as a product of one in itself?
[5:26] That's only half the definition.
[5:28] There's a second half of the definition.
[5:32] because the number one is only divisible by one in itself. The second half is a
[5:39] A number is prime if it is divisible only by one and itself.
[5:45] not a prime number. It's part of the definition. Okay so the reason one is not
[5:54] prime is because we define it not to be a prime number. It's in the definition. So
[6:01] why is one not prime? Because by definition. Okay. Which does make it, every
[6:11] number can be written uniquely as a factor of prime numbers to certain
[6:19] powers. Okay, that's an important thing in number theory. But so this is the only
[6:27] prime factorization of the number 24.
[6:42] To factor means to write as a product.
[6:45] Okay, so To factor means what?
[6:50] To write as a product.
[6:53] Okay, so what is a factor?
[6:58] Okay, one is a factor, 24 is a factor. In fact, what are the factors of 24? 1, 2, 3, 4, 6, 8, 12, and 24.
[7:24] So what do all these numbers, these are all factors of 24. So what's it mean to be a factor?
[7:37] There's kind of two different definitions, but basically as far as numbers go,
[7:42] sorry this is the noun, a number is a factor of another number.
[8:02] divides evenly into it.
[8:12] We'll have to say if it divides evenly into it.
[8:21] Another definition is each, let me just call it a thing, in a product is called a factor.
[8:46] So these are a bunch of the, probably most are all the things.
[8:50] Each one of these things is a factor.
[8:53] A factor, a factor, a factor.
[8:56] Everything inside a product called a factor.
[9:00] Or if it divides evenly into it, it's a factor.
[9:03] Okay, once you find the prime factorization of 72
[9:33] I want each kind factor to the correct power.
[9:37] The prime factorization is 2 cubed times 3 squared.
[9:41] Is that right?
[9:43] These factor it, each of the prime factors.
[9:45] What are all the factors of 18x squared?
[9:48] Now that's the prime factor.
[9:49] What are all the factors of 18x squared? Now we got not just a number but polynomial.
[10:07] So we start with the numbers. There are factors of 18. Which are what? 1, 2, 3, 6, 9, 18.
[10:25] Okay, is x a factor? Yeah, I could write this as 18x times x, it's a factor, or in
[10:37] other words 18x squared divided by x is 18x, it divides in it evenly. So x is a
[10:47] factor? Is x squared a factor? Yeah, right here we already got a product. 18 is a
[10:55] factor, x squared is a factor. What about x cubed? No. Okay, 18 x squared divided by
[11:06] x cubed is 18 over x, which does not divide any evenly. We end up with
[11:11] denominator. Okay so not x cubed. What other factors are there? Definitely 18x squared.
[11:23] is 2x a factor? Yeah 3x yeah 6x 9x 18x what about 2x squared yes 3x squared 6x
[11:51] squared, 9x squared, and 18x squared. Those are all the factors of 18x squared.
[12:02] Its prime factorization is 2 times 3 squared times x squared.
[12:23] Two times three squared times x squared. It's that simple. It's one of every prime factor.
[12:32] X is a prime factor to whatever power that's its prime factorization
[12:51] Find this prime factorization
[12:54] We get that.
[12:58] It's just the prime factorization of the coefficient times the rest of the variables.
[13:09] Slightly new topic.
[13:13] find the common sorry

## 13:37 — Greatest Common Factors

[13:37] What are the common factors of 20 and 24?
[13:43] or let me list the factor to 20.
[13:50] One, two, five groups for.
[14:14] So, the common factors are...
[14:36] I forgot four here.
[14:44] The common factors are one, two, and four. The greatest common factor is four.
[15:04] largest of all the common factors. Okay.
[15:13] 20 is 2 to the second times 5. 24 is 2 cubed times 3.
[15:24] Are those the two prime factorizations?
[15:29] Okay, and what acronym do we use for the greatest common factor?
[15:35] GCF.
[15:36] How can we use the prime factorizations to find the GCF?
[15:41] I
[16:02] How can we use the two prime factorizations to find the greatest common factor?
[16:09] Use every common prime factor to the lowest power.
[16:20] Use every common prime factor to the lowest power.
[16:27] Okay, and then how do I know?
[16:30] Is it 2 squared or 2 cubed or something else?
[16:36] Yeah, it's going to be the lowest of all the exponents.
[16:41] so it's going to be 2 squared.
[16:46] Okay, you could also think of this as
[16:52] 2 times 3 to the 0 times 5 to the 1.
[17:02] Okay, so in fact it's one of every factor to the lowest power, so the two, between the two and the three is 2 squared.
[17:11] between 0 and 1 it's 0, which is just the number 1, and between 5 to the 1 and 5 to the 0,
[17:19] it's times 5 to the 0, which is just 2 squared. So it's one of every common prime factor to
[17:30] the lower of all the values, because 2 cubed does not divide evenly into this, but 2 squared
[17:37] device in both.
[17:38] Now we've got some variables in there.
[17:45] So if we quickly find the prime factorization.
[17:48] It's the same thing but now we have variables in there as well.
[17:52] So between 2 cubed and 2 to the first, the smaller is 2 to the first.
[18:00] Between 3 squared and 3 squared.
[18:03] It's 3 squared.
[18:07] Between x cubed and x to the first we need a x to the first, y squared, y to the fourth, y squared,
[18:17] and there's no common Z so that'd be at 9 the 18 x y squared

## 18:51 — Factor Out the GCF

[18:52] Okay, so what do we call this polynomial?
[18:57] Fourth degree trinomial.
[18:59] We're going to factor out the greatest common factor.
[19:03] So we look for the greatest common factor between three expressions or three
[19:08] monomials.
[19:10] We start the numbers 3, 12, and 18.
[19:13] We really don't even have to start factoring them. What's the largest number that goes into all three of them?
[19:19] Between x to the fourth, x cubed and x squared?
[19:23] x squared goes in all of them.
[19:25] That's the greatest common factor. You put parentheses.
[19:33] 3x squared times what gives 3x to the fourth? x squared.
[19:34] gives you 3x to the fourth? x squared.
[19:40] 3x squared times what gives you negative 12x cubed?
[19:48] 3x squared times what gives you 18x squared?
[19:51] Six.
[19:56] And that's it.
[19:59] The result is 3x squared times the quantity x squared minus 4x plus 6.
[20:02] You find the greatest common factor between each of the terms, bring it on the outside,
[20:09] and then determine what we need to multiply to get that.
[20:12] In fact, that should be the results on our very first problem.
[20:16] Is that right there?
[20:17] Let's try another one.
[21:10] Between 4, 8, and 18, the numerical GCF is 2.
[21:17] Between x cubed, x squared and x to the 4th, the greatest common is x squared.
[21:23] y to the first, y to the fourth, y squared.
[21:29] y to the first
[21:31] is the smallest of the exponents.
[21:38] And then
[21:39] 2x squared y times what gives us this?
[21:46] x
[21:49] 2x squared y times what gives us this?
[21:59] 2x squared y times what gives us this?
[22:09] We factored out the greatest common factor.
[22:16] I could factor out negative 2 instead, but that would not be the greatest common factor.
[22:25] But why is that not the greatest common factor?
[22:28] Because negative 2 is
[22:31] Less than positive. So we, when we're just factoring out the greatest common factor, we never factor out a negative.
[22:40] Sometimes we need to factor out a negative. That will be important when grouping.
[22:44] But for right now, the greatest is always a positive factor.
[22:51] And that's it.
[22:53] Okay, kind of a new topic.
[23:13] Okay, going back to multiplying where we
[23:15] We talked about this the last couple of days. What property do we use? The distributive property.
[23:22] properties. I'm gonna do it the long way, not our shortcut. I want you to write this
[23:29] out and you'll see why in a minute. I'm gonna first take this, distribute the
[23:34] whole thing, each of these. Then we use the distribute property again. Again.
[23:57] And then we typically write it in descending order.
[24:23] Okay, that's what we did the other day.
[24:33] Okay, next.

## 24:36 — Factoring by Grouping

[24:37] I'm just going to say factor.
[24:46] We look at the four terms.
[24:48] What's the greatest common factor between those four terms?
[24:57] Yeah, one. So can we, we could factor out a one, we're not going to.
[25:05] What's it mean to factor?
[25:10] To factor means to write as a product.
[25:11] to write as a product.
[25:12] Besides multiplying by one, is it possible to write this polynomial as a product?
[25:25] We want to rewrite the four-term polynomial as a product of two binomials.
[25:38] We know it factors because we started with two binomials and multiplied them.
[25:48] started with it factored already. We multiplied it out. So how in the world
[26:09] The first group has a greatest common factor of 3x.
[26:19] The second group has a greatest common factor of negative 4.
[26:22] Negative four.
[26:24] and I factor it out. And then once I get it here, what do we have? We really only
[26:31] have one term, this plus this, but we have a common binomial factor. If we have
[26:51] for 3x minus 4.
[26:57] This is called factor by grouping.
[27:11] Can only be used when they're
[27:21] are four terms.
[27:30] Well, it could be six, but that gets complicated.
[27:37] It could be eight terms, but yeah, we're only we're only going to do it when there's four terms.
[27:42] The first step is always factor out the greatest common factor first.
[27:47] Side note, always factor the GCF first. But here the GCF is one, so we're not going to do that.
[28:03] Split the four terms into two groups: the first two terms plus the second two terms.
[28:14] And then within each group, we factor out the greatest common factor.
[28:19] So between 3x cubed and negative 4x squared, we factor an x squared out.
[28:29] 3x minus 4.
[28:34] What's the greatest common factor between 6x and negative 8?
[28:37] 2.
[28:46] Now we have two groups within each group.
[28:50] There's a binomial factor and
[28:54] it's a common binomial factor and so I
[28:59] factor that out.
[29:12] And when I factor it out of here, I'm left with the x squared.
[29:16] This plus is this plus.
[29:19] When I factor out of here, I'm left with the 2.
[29:25] So it's x squared plus 2 times 3x minus 4.
[29:32] And there it is.
[29:33] We factored that.
[29:35] a third degree or a cubic polynomial which has four terms into the product of two binomials.
[29:47] Okay, so we split up into two groups, factor out the GCF out of each group, and then check
[29:53] is there a common binomial factor, if so factor that out. Okay, why on this one are common
[30:03] binomial was 3x minus 4, but when we originally multiplied it, the common
[30:09] binomial was this. Why did that happen? It's because this polynomial with four
[30:23] terms was not written in descending order. If I took this and factor by
[30:31] grouping you will have a common x squared plus 2 with these two switched
[30:37] okay so but that was just side note let's try another one
[30:42] Actually, sorry.
[30:43] Let me change this.
[31:00] Now factor completely. First identify the GCF of all four terms.

## 31:08 — Factor Completely: GCF Then Group

[31:09] Okay, so the factor completely.
[31:16] The first thing we always do is what?
[31:19] Find the GCF and factor up the GCF.
[31:22] What is the GCF?
[31:29] Yeah, I think it's only 10.
[31:31] Factor out the common factor of 10.
[31:49] Now this has four terms.
[31:51] How are we going to do it?
[31:53] Factor by grouping.
[31:55] This 10 outside we just leave out here. Okay so I group into two groups this plus
[32:06] this. So between these two terms what's the greatest common factor? 3x squared.
[32:28] Between 12x and negative 16, what's the greatest common factor?
[32:42] And sure enough, we have a common binomial factor.
[32:49] Let's bring the factor that out.
[33:06] Let's put factor that out.
[33:10] and we're left with 3x squared plus 4. We usually put the coefficients of the
[33:31] monomial in front, but the order of the two binomials does not matter.
[33:36] That's fair.

## 34:15 — A Negative Factor in a Group

[34:16] Find the GCF of the first group and the GCF of the second group.
[34:33] So the greatest common factor is?
[34:36] Two does not divide 15, so the four terms have no numerical GCF greater than one.
[34:42] So we don't factor out a one.
[34:46] We have four terms.
[34:48] Use factoring by grouping.
[34:54] First off, I can't write it like that.
[34:56] Why not? Because I changed that to multiplication.
[35:02] I can't
[35:03] do this. Why not?
[35:08] Okay, yet that negative
[35:11] is only with the 18. So, if there's a
[35:15] subtraction between the third term, we just have to put a little plus and then
[35:21] make sure we get the parentheses correct.
[35:32] And then we factor out the GCF out of here.
[35:38] Not just five.
[35:42] X squared.
[36:04] Factor out negative 6 so both groups contain the same binomial.
[36:08] It is very important.
[36:10] Factoring out positive 6 would produce the opposite binomial, so it would not match.
[36:20] Instead, factor out negative 6 so both groups contain 3x plus 8.
[36:39] Plus 8. Now we have a common binomial factor.
[36:54] Factoring by grouping works when four terms can be grouped to create a common binomial.
[37:02] because they have four terms doesn't mean it's going to work. Not all polynomials are
[37:07] able to be factored. Okay, but if it is able to factor, for example, if this were not,
[37:14] say this were 96 instead this would become well 16 and these would not be
[37:32] common so there'd be no way to factor that out so only sometimes it works it
[37:39] It has to work out nicely.
[37:43] Anyways, there are four. Let's do one more. If you get it right,
[37:48] we'll start homework.

## 37:48 — Final Four-Term Practice

[37:48] Try this final example. Start with the GCF.
[37:52] Factor out the GCF first.
[37:55] Check your result with a neighbor.
[37:56] We're seeing it got the same thing.
[37:57] On Thursday, we will learn to factor polynomials with three terms, called trinomials,
[38:05] and polynomials with two terms, called binomials.