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## Get ready for AP® Calculus

### Course: Get ready for AP® Calculus > Unit 1

Lesson 5: Introduction to factoring# Intro to factors & divisibility

Sal explains what it means for a polynomial to be

*of another polynomial, and what it means for a polynomial to be***a factor***another polynomial.***divisible by**## Want to join the conversation?

- But how did he get 10x in the first place? In the first problem, he only multiplied the similar terms (3 times -2, x times x etc.) But in the second one he multiplied everything(13 votes)
- When multiplying binomials, think of it as doing the distributive property. Multiply each term by each term. So x * x = x^2, while 3 * 7 = 21. But, x * 7 =7x, while 3 * x = 3x. So, x^2 +7x + 3x + 21. Simplifying that, you add the 3x to the 7x to equal 10x. The final answer is, x^2 + 10x + 21(6 votes)

- Hi,

Just to get some clarity,

What is the difference between binomial, polynomial, trinomial, etc?(5 votes)- A monomial is a polynomial with 1 term.

A binomial is a polynomial with 2 terms.

A trinomial is a polynomial with 3 terms.(25 votes)

- can decimals be factors or is it just integers? as an example, you can definitely say that 3 is a factor of 6, but can you say that 2.5 is a factor of 5?(7 votes)
- Great question! When we talk about factors of whole numbers, we are looking for whole numbers. 2.5 is not a whole number, so it is not a factor of 5. Negative integers, like -1 and -5, are not factors either, because they are not whole numbers. The only factors of 5 are 1 and 5, making 5 prime.(8 votes)

- Why did he put 10x at4:15??(1 vote)
- When you multiply (x+3)(x+7), you would get x squared plus 7x plus 3x plus 21. You have 3x and 7x, which add together to get 10x.(8 votes)

- What exactly is the purpose of factorizing something?(4 votes)
- @LestathRiveraD

Great Question! Similarly in algebra, factoring is a remarkably powerful tool, which is used at every level. It provides a standard method for solving quadratic equations as well, of course, as for simplifying complicated expressions. It is also useful when graphing functions. Factoring (or factorising) is the opposite of expanding.

This is a great process of simplification. Also, factoring is a complementary operation to the distributive property, it is a way to “unpack” the multiplication done by applying the distributive property. Reorganizing polynomials by factoring allows us to find solutions for certain types of polynomials.

Hope this helps.(1 vote)

- what is the difference between polynomials, and binomials?(3 votes)
- A polynomial is simply a fancy word to describe a math statement with several terms. A mononomial is the opposite - it is a statement with only one term. A binomial is a type of polynomial that contains exactly two terms. All of these are pretty simple to remember because they use the same Greek/Latin roots that many other math terms use. (Poly means many, mono means one, and bi means two). Here are some examples of all of these concepts.

Polynomial:

- 4x^2 + 5x + 6

- x + a + c

- 433 + 19b + 17c + 25a

Mononomial:

- 5x

- a

- 19

- 7

Binomial:

- x + 7

- 4x + 9

- 9a + b

Hope that helps!(2 votes)

- we know that

(a+b)(a-b)=a^2-b^2

This means that,

a+b is a factor of a^2-b^2, ALSO

a^2-b^2 is divisible by a+b

but dividing a^2-b^2 by a+b does not give a-b

plz help me to understand this concept in term of this formula

THANK YOU.(1 vote)- Look, if a=1&b=2, a^2-b^2=-3, and -3/a+b=-3/3=-1, a-b=-1, so (a^2-b^2)/(a+b)=a-b. (Replace unknowns with numbers)(5 votes)

- If x^2 + 10x + 21 is a trinomial, then is x^2 + 3x + 7x + 21 a quadrinomial, or still a trinomial, since it can be simplified to the former?(2 votes)
- In its expanded form, it is a quadrinomial because it has 4 terms at that point. However, it is still a quadratic expression.(3 votes)

- Is not (3xy)(-2x^2y^3) the same as [(xy) + (xy) + (xy)][(-x^2y^3) + (-x^2y^3)] ?(2 votes)
- Yes, but why do you want / need to write the expression that way? This is rarely done.(2 votes)

- Would 1/2 and 20 be factors of 10? Could fractions and decimals be defined as factors? It doesn't specify if the factors must be whole numbers or integers. Also, if you have 3xy for example, and you say that 6xy^2 is a factor of it, would that be wrong? If so, is it because a monomial cannot have a fractional coefficient or is it because factors cannot be decimals or fractions?(1 vote)
- Factors are integers that will divide evenly into a number without leaving a remainder. 1/2 would not be considered a factor of 10 as it is not an integer. 20 would not be a factor of 10 because it does not divide evenly into 10.(4 votes)

## Video transcript

- [Voiceover] You're probably
familiar with the general term factor. So if I were to say: What
are the factors of 12, you could say: Well what
are the whole numbers that I can multiply by another
whole number to get 12? So for examples, you
could say things like, well I could multiply
one times 12 to get 12. So you could say that
one is a factor of 12. You could even say that
12 is a factor of 12. You could say two times
six is equal to 12, so you could say that
two is a factor of 12 and that six is also a factor of 12. And of course three times four is also 12, so both three and four are factors of 12. So if you said well what
are the factors of 12 and you've seen this
before, well you could say: one, two, three, four, six, and 12, those are all factors of 12. And you could also phase
it the other way around, so let me just give an example. So if I were to pick on
three, I could say that three is a factor of 12. Or to phrase it slightly
differently, I could say that 12 is divisible, 12 is divisible by three. Now what I wanna do in this
video is extend this idea of being a factor or divisibility
into the algebraic world. So for example, if I were to take 3xy. So this is a monomial with
an integer coefficient. Three is an integer right over here. And if I were to multiply
it with another monomial with an integer coefficient, I don't know, let's say times negative two X squared, Y to the third power, what is this going to be equal to? Well this would be equal to, if we multiply the coefficients,
three times negative two is going to be negative six. X times X squared is X to the third power. And then Y times Y to the third is Y to the fourth power. And so what we could say is,
if we wanted to say factors of negative six X to the
third, Y to the fourth, we could say that 3xy is a factor of this just as an example; so
let me write that down. We could write that 3xy is a factor of, is a factor of... of negative six X to the
third power, Y to the fourth, or we could phrase that
the other way around. We could say that negative
six X to the third, Y to the fourth, is divisible by, is divisible by 3xy. So hopefully you're seeing the parallels. If I'm taking these two monomials
with integer coefficients and I multiply 'em and I get
this other, in this case, this other monomial, I could
say that either one of these and there's actually
other factors of this, but I could say either
one of these is a factor of this monomial, or we
could say that negative six X to the third, Y to the four is divisible by one of its factors. And we could even extend this
to binomials or polynomials. For example, if I were to take, if I were to take, let me scroll down a little bit, whoops, if I were to take, let me say X plus three and I wanted to multiply it times X plus seven, we know that this is going to be equal to, if I were to write it as a trinomial, it's gonna be X times X, so X squared, and then it's gonna be
three X plus seven X, so plus 10x; and if any
of this looks familiar, we have a lot of videos
where we go in detail of multiplying binomials like this. And then three times seven is 21. Plus 21. So because I multiplied these
two, in this case binomials, or we could consider
themselves to be polynomials, polynomials or binomials
with integer coefficients. Notice the coefficients
here, they're one, one. The constants here, they're all integers. Because I'm dealing with all
integers here, we could say that either one of these
binomials is a factor of this trinomial, or we could say this
trinomial is divisible by either one of these. So let me write that down. So I could say, I'll just
pick on X plus seven. We could say that X plus seven is a factor, is a factor of X squared plus 10x plus 21; or we could say that X squared plus 10x plus 21 is divisible by, is divisible by I could say X plus three
or I could say X plus seven is divisible by, X plus seven. And the key is, is that
both of these binomials, or even if we were
dealing with polynomials, we are dealing with
things that have integer, we're dealing with things that
have integer coefficients.