Integrated Math 2–3 Honors — Polynomial Vocabulary, Classification, and Operations
Corrected lesson transcript


## 0:00 — Polynomial Vocabulary and Operations


## 0:05 — What Is a Monomial?

[0:05] Alright, we're talking about polynomials.
[0:14] We're going to start out with the definition.
[0:32] A monomial is a number, a variable, or the product of numbers and variables raised to
[0:42] whole number powers.
[0:43] Let's see if this gets...
[1:04] A monomial is a number, a variable, or the product of numbers and variables raised to
[1:13] poll number.
[1:50] Uh, examples.
[1:57] Is three a monomial?
[2:02] Yes.
[2:03] Yes, why?
[2:03] Because three is a number. That's it. Simple. Is x a monomial? Yes. Because it's a variable.
[2:24] 2.
[2:28] Is 4x squared a monomial?
[2:34] Yes.
[2:35] Yes, why?
[2:37] Product of numbers and variables raised to whole numbers.
[2:41] Product of numbers and variables raised to whole numbers.
[2:45] Powers.
[2:47] OK, not monomials.
[3:04] Is x to the negative 1 same thing as 1 over x?
[3:12] Is that a monolion?
[3:14] No.
[3:15] No, why not?
[3:16] It's nominal.
[3:19] Because?
[3:22] It's not raised to a whole number.
[3:24] number power.
[3:27] Negative 1 is not a whole number.
[3:30] OK, what are the whole numbers?
[3:33] Positive 2.
[3:36] So the 1, 2, 3, 4, and so on, the counting number
[3:41] is also known as the natural numbers.
[3:45] You add 0 to it, 0, 1, 2, 3, 4, and so on.
[3:48] Those are the whole number, set of whole numbers.
[3:54] the square root of X is that a monomial why not same thing as X to the 1 half
[4:05] which is not a whole number X squared plus 4x is that a monomial it's not
[4:21] Not because?
[4:26] It's not a product of numbers and variables.
[4:28] We have a sum here.
[4:29] This is not a monomial.
[4:32] Okay.
[4:38] What about the square root of 5?
[4:39] Is that a monomial?
[4:44] Same thing as 5 to what power?
[4:47] 1 half power.
[4:49] Is it a monomial or no?
[4:50] No.
[4:53] Is the square root of 5 a number?
[4:56] Yeah.
[4:57] Yeah, it's a number.
[4:58] It's definitely a number.
[5:01] Okay, it is a monomial.
[5:03] Only the variables has to be raised to a whole number of powers.
[5:08] Oh, I noticed you put it in each example instead of five.
[5:12] And yes, I put it under the example, not the not a monomial.
[5:16] That's the other clue.
[5:17] OK, next definition.
[5:22] Actually, let's do one more, it's not...
[5:27] 2 to the power of x. Is that a monomial?
[5:31] No. No, why not?
[5:33] Because the exponent's a monomial.
[5:37] Yeah, the variable's in the exponent. OK.
[5:42] We can't have any variables in the exponent.
[5:45] It's not a monomial.
[5:47] Next definition.
[5:56] A polynomial is a monomial for the sum of monomials.

## 6:07 — Build Polynomials from Monomials

[6:51] Is this a polynomial?
[6:54] Yes. Yes, why?
[6:56] Because it's a monomial.
[7:06] Is this a polynomial?
[7:08] Yes, yes, why it's the sum of three monomials
[7:23] Is this a polynomial
[7:30] For those who said no, why not
[7:33] It's not
[7:36] Is this the same thing
[7:42] Is negative 8x squared a monomial I was not I was trying to trick you
[7:50] But make sure we get the right definition
[7:55] This is a monomial sorry, this is a polynomial
[7:59] even though there's a subtraction,
[8:01] we always think that's just a plus and negative, that number.
[8:05] OK.
[8:07] Next definition.
[8:12] The number in a number being multiplied, not the exponent.
[8:27] Monomial is called the coefficient. Okay, the number, not the exponent.
[9:02] what's the coefficient here? 5. the coefficient here? 3. what is the
[9:12] here just it's very important it is negative 8 not 8 okay yeah hey Oh a term

## 9:48 — Terms, Coefficients, and Names

[10:35] How many terms does this have?
[10:40] Because we think of this as 5x to the fourth plus negative 7x squared plus 8.
[10:47] It has three terms. One, two, three terms.
[11:03] A polynomial with two terms is called a binomial.
[11:23] A polynomial with two terms is called a binomial.
[11:27] A polynomial with one term is called a monomial.
[11:31] two-term binomial a polynomial with three terms is called a trinomial a
[11:53] polynomial with four terms could be called a quadrinal meal but we don't
[12:11] We just call it a called a polynomial with four terms.
[12:20] Oh wow, that's pretty elaborate.
[12:28] Once you get past three, four, and on, we call it a polynomial with n terms.
[12:36] What's that?
[12:36] What is that commutation line?
[12:37] That means repeat above a polynomial.
[12:43] A polynomial with four terms is called a polynomial with four terms.
[12:47] That's right.
[12:48] A polynomial with seven terms is called?
[12:50] A polynomial with seven terms.
[12:51] A polynomial with seven terms.
[12:53] So we've got a monomial, a binomial.
[12:55] Let me do a couple quick examples, right?
[12:58] For example, 3x cubed plus 4x is a binomial.
[13:06] Why? Because it has two terms. Is it also a polynomial? Yeah, definitely. Okay, three terms,
[13:16] X squared plus 2x minus 5 will be a trinomial
[13:34] This would be called a
[13:36] Polynomial with four terms so monomial binomial trinomial polynomial four terms polynomial five terms and so on
[13:50] okay next definition the degree of a polynomial sorry of a monomial the sum
[14:12] of the exponents.

## 14:14 — Degree of a Term and a Polynomial

[14:27] exponents
[14:42] So samples.
[14:50] What is the degree?
[14:53] 6.
[15:05] What is the degree?
[15:08] 2.
[15:14] What is the degree here?
[15:18] 1. 1. Is the number 7 a monolu?
[15:25] Yeah. It is. What's the degree?
[15:28] 1. I don't see any variables. This is degree 1.
[15:35] Oh, it's going to be 3. Ok, we could think of this as 7, same thing as 7x to the 0 power.
[15:43] what is X and zero power one so what is the degree zero okay we're not going to
[15:53] deal with this a lot in this class but 7x squared y to the fourth what is the
[16:00] degree it's the sum of all the exponents of the variables so 2 plus 4 is 6
[16:12] Okay, every number has degree zero. We call this a constant. Okay, next definition.
[16:31] the degree of a polynomial not a monomial
[16:58] is the degree of the term with the largest degree.
[17:40] How many terms does this have?
[17:43] Three.
[17:44] What's the degree of this term?
[17:47] Two.
[17:47] One.
[17:48] One.
[17:49] Degree of this term?
[17:50] Two.
[17:51] Two.
[17:51] Degree of this term?
[17:52] Zero.
[17:53] Zero.
[17:54] The degree of the whole polynomial is?
[17:56] Two.
[17:56] Two.
[17:59] Because it's the degree of this term, which is two.
[18:30] What is the degree of this term?
[18:33] Degree of this term?
[18:35] Degree of this term?
[18:36] Four.
[18:37] Degree of this term?
[18:38] Five.
[18:39] The term with the largest is this, so the degree of the whole polynomial is?
[18:43] Seven.
[18:45] Seven.
[18:50] Another definition, the term with the largest degree is called the leading term.

## 19:19 — Leading Term and Leading Coefficient

[19:20] Even though it doesn't have to be in front, it is the leading term.
[19:32] What is the leading term in this?
[19:35] 7x to the 7.
[19:38] Negative 7x squared.
[19:45] What is the leading term here?
[19:49] 5x to the 7.
[19:51] 5x to the 7.
[19:56] The coefficient of the leading term is called... anybody want to take a guess?
[20:14] Leading coefficient.
[20:16] Leading coefficients.
[20:54] What is the leading term?
[20:58] X cubed.
[20:59] What is the leading coefficient?
[21:02] 1.
[21:14] What is the leading coefficient here?
[21:21] negative 7
[21:38] okay now we got some names
[21:40] of polynomials with different degrees
[21:44] so the degree the name and then I'll give examples the degree zero it's

## 21:51 — Polynomial Names by Degree

[22:11] It's called a constant polynomial.
[22:23] Examples would be like 5 or negative pi.
[22:32] The constant doesn't change.
[22:34] degree one are called linear polynomials.
[22:48] Three x plus four, negative eight x.
[23:03] degree two are called quadratic polynomials.
[23:23] Examples would be x squared.
[23:28] Three x squared plus four x minus five.
[23:37] Two x squared minus eight x.
[23:42] Long as the degree is two,
[23:45] it's called a quadratic polynomial.
[23:51] what does quad usually mean? 4. that's not what it means here it has to do with the Latin word meaning
[24:00] square okay right because the exponent is second power which is a square right we even say x squared
[24:13] so don't confuse that degree three
[24:28] are called, you want to take a guess?
[24:33] Cubic polynomials.
[24:42] Examples are x cubed is a cubic polynomial. 4x cubed minus x squared.
[25:03] Anytime the degree three, the cubic polynomial.
[25:09] We have a couple of other names.
[25:13] degree four is called
[25:19] cork tick x to the fourth minus x cubed or so on.
[25:40] Often after cubic, we often just say polynomial of degree 4.
[26:00] 5 is also known as quintic polynomial.
[26:09] But again, after 3, we usually just say a polynomial of degree 5.
[26:23] And we can keep going.
[26:24] n. Any degree n, we just say a polynomial of degree n. Like 5x to the 8th power is a
[26:46] n. Any degree n, we just say a polynomial of degree n. Like 5x to the 8th power is a

## 26:51 — Classify by Degree and Terms

[26:56] Okay, the degree of the polynomial is the most important thing about the polynomial.
[27:04] Whether it has one term, two terms, three terms, and so on, that's a lot less important.
[27:11] But we will classify them by both degree and the name.
[27:15] Okay, so we're gonna classify by degree and name.
[27:48] What is the degree here?
[27:51] So the degree is 2, we call that a quadratic polynomial.
[28:05] Is it a monomial, binomial, trinomial? It's a trinomial.
[28:11] So when we call this a trinomial, you replace the word polynomial.
[28:18] So we would call this a quadratic trinomial.
[28:26] So any time you use the word monomial or binomial or trinomial, we just replace the word polynomial
[28:33] with that term.
[28:41] What would this be called?
[28:49] First off, what's the degree?
[28:53] What's the leading term?
[28:56] 7x.
[28:56] The leading term would be 7x.
[28:59] The leading coefficient would be 7.
[29:01] OK, so since the degree is 1, it's called a linear polynomial.
[29:11] And then we replace the word polynomial
[29:13] with linear binomial.
[29:28] The number 8 would be of degree 0, that's x to the 0th power, 0.
[29:37] we call zero degree polynomials constant
[29:49] polynomial but we replace the word polynomial with monomial in fact all zero
[30:03] degree polynomials are constant and there's going to only be one term so all
[30:11] constant polynomials are constant monomials linear we could have either a
[30:17] constant binomial or a constant monomial okay
[30:24] Uh, let's do a few more.
[30:27] I think we're going to come up with this one.
[30:31] Except for the last apartment.
[30:34] There's a lot of x in there.
[30:38] There's a lot of x in there.
[30:44] Plus two, let's make it one.
[30:48] What is the degree here?
[30:49] 6. 6. OK. So we just say a 6 degree polynomial. OK. Once we get past three terms, we stop
[31:05] calling it a monomial, binomial, trinomial. So we say 6 degree polynomial with four terms.
[31:16] Shh.
[31:46] What is the degree of this polynomial?
[31:50] Tenth degree. Okay, and there's only two terms we call it a tenth degree
[31:59] Binomial
[32:07] Okay
[32:08] Again, the degree is the most important thing
[32:11] however many terms
[32:13] Is less important and if you use a monomial binomial trying to use to replace the word polynomial
[32:21] Let's take we're gonna pause we still got a lot more to learn

## 32:26 — Adding and Subtracting Polynomials

[32:26] But we're gonna practice these
[32:29] Let me pause
[32:35] Now we're talking about adding and subtracting polynomials, but you've probably done in the past
[32:41] But let's just review
[33:13] So we're adding two quadratic polynomials to quadratic trinomials to be precise.
[33:24] Do we need these parentheses here?
[33:27] No, right, because this addition is the same as this addition, same as this.
[33:42] So, and how are we going to add these?
[33:44] We've done this in the past I believe.
[33:47] We're going to add like terms.
[33:48] We could rearrange it.
[33:51] You don't need to do this.
[33:52] we usually do this part in our head.
[34:02] Okay and what are like terms?
[34:09] Terms with the same degree, okay, are like terms. So they have to have the same variables
[34:18] raised to the same powers but these only have the X's in and how are we going to add like terms?
[34:32] 1x squared plus 4x squared is 5x squared. You could factor out an x squared we get 1 plus 4
[34:41] it's really because of the distributive property but again we do this part in our head.
[34:45] But same thing here, these are like terms.
[34:55] And we get, what is that, negative 5x.
[34:59] And the negative 5 to 6 is 1.
[35:02] And we usually don't write plus and negative,
[35:05] so we just rewrite as 5x squared minus 5x plus 1.
[35:11] OK.
[35:11] And typically we write it in, it's called descending order.
[35:19] Descending from the term with the largest degree
[35:22] down to the term with the smallest degree.
[35:24] So we've got a quadratic, a linear, a constant monomial.
[35:28] So we write it in descending order.
[35:54] I
[35:55] Will subtracting polynomials
[36:01] First up do I need these parentheses here, or can I just remove them?
[36:08] Definitely not.
[36:10] Because we're subtracting this entire thing.
[36:13] Not subtract.
[36:14] Without this, it would just be a minus 5x squared, which is not the case.
[36:19] So this is very important.
[36:21] We need to subtract, we add the opposite, and the opposite is we just distribute the
[36:28] negative is what I call it, even though technically it's not that.
[36:44] Okay, so you add the opposite and you just flip all the signs of the whatever is after
[36:50] this fraction or you just think of it as distributing the negative or the negative one.
[36:59] Does X cubed have any like terms?
[37:01] nope 4x negative 4x squared plus negative 5x squared is negative 9x
[37:11] squared x plus negative 6x is negative 5x and minus 7 that's it so we subtract
[37:27] distribute that negative or we add the opposite of a polynomial which is all

## 37:30 — Multiply by a Monomial

[37:32] the flip signs not too complicated right okay you've done that in the past okay
[37:42] so that should be a review now let's talk about multiplying polynomials
[37:57] I think I know where it is.
[38:07] So here we're multiplying a monomial times a trinomial.
[38:13] We're multiplying one quadratic times another quadratic.
[38:17] And what property do we have to use?
[38:20] The distributive property of multiplication over a distribution.
[38:23] Very good. The distributive property of multiplication, this multiplication.
[38:27] over addition we simply distribute just like we did with radicals the other day
[38:39] Two, let's draw.
[38:47] Three.
[38:48] Four.
[38:50] Six.
[38:51] Seven.
[38:51] Eight.
[38:52] Ten.
[38:53] Ten.
[38:54] And then we just use our rules of x on it that we've done.
[38:58] Three x squared times x squared.
[39:01] X to the two plus two.
[39:04] X to the fourth.
[39:09] minus 12.
[39:13] x squared times x is x to the 2 plus 1, which is x cubed.
[39:25] 15x squared.
[39:30] It's just the rules of exponents.
[39:34] And we end up with this.
[39:38] What is the degree here?
[39:41] Four.
[39:42] Four.
[39:43] How many terms?
[39:45] Three.
[39:46] Do we call this a?
[39:48] Four degrees.
[39:49] He's trying to rattle me.
[40:11] Go ahead and distribute.
[40:18] you.
[41:14] 42x to the fourth.
[41:19] What about that?
[41:25] Did we all get that? Yes, sir. Yeah.
[41:28] Multiplying is all about just using the distributive property.
[41:32] Because polynomials have multiple terms.

## 41:34 — Multiply Two Binomials

[41:36] Now, so that's multiplying monomials
[41:40] times any polynomial you just distribute the monomial to every term and then
[41:46] simplify now let's talk about a binomial times a binomial is a linear binomial
[42:05] The linear binomial what property we have to use
[42:14] The distributive property of multiplication over
[42:18] addition
[42:19] Just like we use radicals. We have to distribute the whole
[42:24] Binomial the whole polynomial to every term in the other
[42:32] We get 2x plus 3
[42:34] times x plus 2x plus 3 times negative 5.
[42:50] And now what do we have to do?
[42:53] Use the distributive property two more times.
[42:59] Just like we did with radicals.
[43:28] And then the last step.
[43:32] Find like terms. Find like terms. I only have two like terms here. So it's 2x squared 3
[43:41] minus 10 minus 7x. Okay is there a shortcut where we could just bypass this
[44:02] whole step. Why don't you talk to your neighbor, what's a shortcut?
[44:07] it.
[44:25] Uh, anybody got a shortcut?

## 44:36 — Multiply Larger Polynomials

[44:37] No shortcuts?
[44:40] Okay, let's just do the long cut from that one. That's fine. I like it. Let's try another one.
[45:01] I don't know.
[45:02] Can I spell it?
[45:03] Go.
[45:05] They can't.
[45:06] They can't.
[45:07] They can't.
[45:08] They can't.
[45:08] They can't.
[45:11] So what do we do?
[45:15] We distribute.
[45:18] OK.
[45:19] Does it matter if I distribute this here
[45:22] or if I distribute this here?
[45:26] It doesn't matter.
[45:27] It works out at the end the same.
[45:32] Let's just do that. I always go left to right.
[45:57] So we distribute the 4x cubed minus 3x squared.
[46:03] and now what do we gotta do?
[46:06] distribute again
[46:10] I wish there was an easier way
[46:12] In what order do we write our final answering?
[46:14] What's it called descending descending the term with the largest degree down to the term of a degree?
[46:22] So let's go back
[46:25] Is there a shortcut where we can skip this entire step, yes, what's the shortcut?
[46:32] aluminum foil method
[46:34] and
[46:35] well, five
[46:37] why are you guys talking about aluminum foil?
[46:41] I think like you multiply the front
[46:43] two terms together
[46:44] then you multiply the outside terms
[46:47] and then you multiply the inside terms
[46:50] and the last terms
[46:52] okay, I like that
[46:55] except what happens when we have
[47:06] Two terms times three terms.
[47:19] Okay so let's do a shortcut without any weird words like foil.
[47:31] Let's look at this. Where did these two terms come from?
[47:39] Was this times this. Okay. Which let me erase this.
[47:54] so it's this term times this term where did this the negative 3x squared and the
[48:02] 2x squared come from this and then the 4x cubed and the 5x here and the negative 3x
[48:18] squared and the 5x there okay so we can skip this entire step by just taking
[48:28] each term and distribute it to the other terms in the other polynomial and then
[48:38] you go to the next term you distribute to each of the terms and the other one
[48:43] it's just a bunch of distributing we have a binomial times a binomial some
[48:49] people like the acronym foil okay first outside inside last but I don't mess with
[48:57] that because once you get two times three terms a binomial times a trinomial
[49:04] you just take the first term just distribute it to every term and then
[49:13] And you go to the next term, distribute to the next term.
[49:21] We got 2x times x squared plus 2x times 4x plus 2x times negative 5.
[49:36] Those should all be in purple.
[49:43] Oops. Let's try one more time. There we go. And then the second term times the all of
[50:04] 3 times x squared plus 3 times 4x plus 3 times maybe 5.
[50:14] Eventually we even do this in our head and we start writing all 6 terms.
[50:17] So 2 terms times 3 terms, 2 times 3, we always get that number of terms.
[50:22] 2x cubed plus 8x squared minus 10x 3x squared plus 12x minus 15.
[50:37] And once we do that, we just combine like terms.
[50:43] This guy likes this term 11x squared.
[50:53] That's two.
[50:57] Minus 15.
[51:02] I think it's a cubic polynomial.
[51:07] Cubic is correct, the degree is three,
[51:11] but the number of terms is four.
[51:13] So we spell it cubic polynomial with four terms.
[51:19] Okay.
[51:22] Try another one.
[51:40] Just take each term in any one of them.
[51:42] I always go left right.
[51:44] We distribute to every term in the other polynomial.
[51:55] Start distributing and then simplifying and then combining like turns.

## 52:43 — Final Polynomial Product

[52:43] A trinomial times a trinomial will have to redistribute how many terms you're going to have.
[52:48] All I have is nine.
[52:50] Nine.
[52:50] Three times three is nine.
[52:52] So write out all nine terms.
[52:55] And then start simplifying.
[53:23] Once we get good at this, you can just multiply in your head.
[53:27] Expert times expert.
[54:24] And then...
[54:25] I'm always like...
[54:26] Still being wrong?
[54:31] Minus 7x...
[54:33] Plus 20, there we go.
[54:34] Oh my goodness.
[54:36] Wait, you didn't like it.
[54:37] You told me it was wrong.
[54:41] I said his was wrong.
[54:42] Why are you looking at the statement from the guy?
[54:44] Are you telling me to do it, bro?
[54:46] Wait a minute.
[54:49] Okay, what questions do we have on this?
[54:53] It's just a lot of distributing
[54:56] using the distributive property of addition,
[55:00] of multiplication over addition.
[55:01] Okay, we're gonna stop here.
[55:03] Yes.