The lesson was a little hard man...

I'm glad I'm not the only one :/ but if we practice a lot we'll get it eventually!

A number that multiplies to another number.

Hey all. Confused on last question. Wouldn’t x2+5x be a factor since you just cancel the x2+5x on top and bottom? Or is it not just because of the problem being about Area?

I realize this was posted 8 months ago, but this is a common mistake so I would like to address it. x^2+5x is not a factor of this expression because it is being added to 4. If that sum were multiplied by 4 instead of added to it, then it would be a factor. The fact that the expression is a sum of x^2+5 and 4 and not a product of the two means that x^2+5 cannot be a factor of x^2+5x+4. I hope that makes sense and clears this up for anyone else wondering the same thing.

What's the easiest way to tell a number is a factor of another?

Think about what 2 numbers *multiply* together to make that number. For example *3 x 4 = 12* therefore *3 and 4* are *factors*.

A monomial is a math statement with exactly one term. A term can be anything with multiplication or division and it can even contain variables, but it cannot contain addition or subtraction. Here are some examples of what is and isn't a monomial. *IS* a monomial - 9x - 14a/2 - 4 - 17ab^2 *IS NOT* a monomial - 14x + 4 - 919 - 7 - x^2 + 4x + 9 An easy way to remember what a monomial is can be to break it down into its Latin/Greek roots. Mono - nomial *mono* means one *nomial* means number or term Hope this helps!

How does this work? I mean I get it but it's really hard. (btw im doing 8th grade math and im in 7th grade.)

(Assuming the denominator is a monomial) First when you are dividing a polynomial, it's better to take each element separated by + or - in the numerator. For example, (2x^2 + 6x)/2x -> 2x^2/2x + 6x/2x This is basically separating a fraction into smaller fractions. Then the next thing you will do is think as each variable / constant that is being multiplied not placed together. Consider 2x^2/2x So you can first take 2/2 and compute it, which results to 1. Then you consider x^2/x, which is x^(2-1) = x. Consider 15x^2y^6 / 10x^4y^3. First take 15 / 10, which is 3 / 2. Then consider x^2 / x^4, which is 1 / x^2. Last y^6 / y^3 which is y^3. The reason why this works is because of the fundamental of algebra. Hope this helps

Does the factor of a polynomial always have to be a monomial? e.g. 3(3x^2 + 3x + 3) factors would be 3 and (3x^2 + 3x + 3)?

A polynomial can have a monorail factor if it's terms have a common factor.But it can also have binomial and trionomial factors that can't be factored further.For example: x^3-1=(x-1)(x^2+x+1) 3(3x^2+3x+3)=9(x^2+x+1) So, the factors are 9 and (x^2+x+1).

Main content

Algebra 1

Course: Algebra 1 > Unit 13

Lesson 4: Introduction to factoring

Intro to factors & divisibility

Google Classroom

Learn what it means for polynomials to be factors of other polynomials or to be divisible by them.

What we need to know for this lesson

A monomial is an expression that is the product of constants and nonnegative integer powers of

x

, like

3 x^{2}

. A polynomial is an expression that consists of a sum of monomials, like

3 x^{2} + 6 x - 1

What we will learn in this lesson

In this lesson, we will explore the relationship between factors and divisibility in polynomials and also learn how to determine if one polynomial is a factor of another.

Factors and divisibility in integers

In general, two integers that multiply to obtain a number are considered factors of that number.

For example, since

14 = 2 \cdot 7

, we know that

2

and

7

are factors of

14

One number is divisible by another number if the result of the division is an integer.

For example, since

\frac{15}{3} = 5

and

\frac{15}{5} = 3

, then

15

is divisible by

3

and

5

. However, since

\frac{9}{4} = 2.25

, then

9

is not divisible by

4

Notice the mutual relationship between factors and divisibility:

Since

14 = 2 \cdot 7

(which means

2

is a factor of

14

), we know that

\frac{14}{2} = 7

(which means

14

is divisible by

2

\underset{2 is a factor of 14}{\underset{⏟}{14 = 2 \cdot 7}} ⟶ \underset{14 is divisible by 2}{\underset{⏟}{\frac{14}{2} = 7}}

In the other direction, since

\frac{15}{3} = 5

(which means

15

is divisible by

3

), we know that

15 = 3 \cdot 5

(which means

3

is a factor of

15

\underset{15 is divisible by 3}{\underset{⏟}{\frac{15}{3} = 5}} ⟶ \underset{3 is a factor of 15}{\underset{⏟}{15 = 3 \cdot 5}}

This is true in general: If

a

is a factor of

b

, then

b

is divisible by

a

, and vice versa.

Factors and divisibility in polynomials

This knowledge can be applied to polynomials as well.

When two or more polynomials are multiplied, we call each of these polynomials factors of the product.

For example, we know that

2 x (x + 3) = 2 x^{2} + 6 x

. This means that

2 x

and

x + 3

are factors of

2 x^{2} + 6 x

Also, one polynomial is divisible by another polynomial if the quotient is also a polynomial.

For example, since

\frac{6 x^{2}}{3 x} = 2 x

and since

\frac{6 x^{2}}{2 x} = 3 x

, then

6 x^{2}

is divisible by

3 x

and

2 x

. However, since

\frac{4 x}{2 x^{2}} = \frac{2}{x}

, we know that

4 x

is not divisible by

2 x^{2}

With polynomials, we can note the same relationship between factors and divisibility as with integers.

\underset{2 x is a factor of 2 x^{2} + 6 x}{\underset{⏟}{2 x \cdot (x + 3) = 2 x^{2} + 6 x}} ⟶ \underset{2 x^{2} + 6 x is divisible by 2 x}{\underset{⏟}{\frac{2 x^{2} + 6 x}{2 x} = x + 3}}

\underset{6 x^{2} is divisible by 3 x}{\underset{⏟}{\frac{6 x^{2}}{3 x} = 2 x}} ⟶ \underset{3 x is a factor of 6 x^{2}}{\underset{⏟}{3 x \cdot (2 x) = 6 x^{2}}}

In general, if

p = q \cdot r

for polynomials

p

q

, and

r

, then we know the following:

$q$ ‍ and $r$ ‍ are factors of $p$ ‍.
$p$ ‍ is divisible by $q$ ‍ and $r$ ‍.

Check your understanding

1) Complete the sentence about the relationship expressed by $3 x (x + 2) = 3 x^{2} + 6 x$ ‍.

x + 2

3 x^{2} + 6 x

, and

3 x^{2} + 6 x

x + 2

[I need help!]

2) A teacher writes the following product on the board:

(3 x^{2}) (4 x) = 12 x^{3}

Miles concludes that

3 x^{2}

is a factor of

12 x^{3}

Jude concludes that

12 x^{3}

is divisible by

4 x

Who is correct?

[I need help!]

Determining factors and divisibility

Example 1: Is $24 x^{4}$ ‍ divisible by $8 x^{3}$ ‍?

To answer this question, we can find and simplify

\frac{24 x^{4}}{8 x^{3}}

. If the result is a monomial, then

24 x^{4}

is divisible by

8 x^{3}

. If the result is not a monomial, then

24 x^{4}

is not divisible by

8 x^{3}

\begin{aligned} \frac{24 x^{4}}{8 x^{3}} & = \frac{24}{8} \cdot \frac{x^{4}}{x^{3}} \\ = 3 \cdot x^{1} & \frac{a^{m}}{a^{n}} = a^{m - n} \\ = 3 x \end{aligned}

Since the result is a monomial, we know that

24 x^{4}

is divisible by

8 x^{3}

. (This also implies that

8 x^{3}

is a factor of

24 x^{4}

Example 2: Is $4 x^{6}$ ‍ a factor of $32 x^{3}$ ‍?

4 x^{6}

is a factor of

32 x^{3}

, then

32 x^{3}

is divisible by

4 x^{6}

. So let's find and simplify

\frac{32 x^{3}}{4 x^{6}}

\begin{aligned} \frac{32 x^{3}}{4 x^{6}} & = \frac{32}{4} \cdot \frac{x^{3}}{x^{6}} \\ = 8 \cdot x^{- 3} & \frac{a^{m}}{a^{n}} = a^{m - n} \\ = 8 \cdot \frac{1}{x^{3}} & a^{- m} = \frac{1}{a^{m}} \\ = \frac{8}{x^{3}} \end{aligned}

Notice that the term

\frac{8}{x^{3}}

is not a monomial since it is a quotient, not a product. Therefore we can conclude that

4 x^{6}

is not a factor of

32 x^{3}

A summary

In general, to determine whether one polynomial

p

is divisible by another polynomial

q

, or equivalently whether

q

is a factor of

p

, we can find and examine

\frac{p (x)}{q (x)}

If the simplified form is a polynomial, then

p

is divisible by

q

and

q

is a factor of

p

Check your understanding

3) Is $30 x^{4}$ ‍ divisible by $2 x^{2}$ ‍?

[I need help!]

4) Is $12 x^{2}$ ‍ a factor of $6 x$ ‍?

[I need help!]

Challenge problems

5*) Which of the following monomials are factors of $15 x^{2} y^{6}$ ‍ ?

	Factor	Not Factor
$3 x^{2} y^{5}$ ‍
$5 x$ ‍
$10 x^{4} y^{3}$ ‍

[I need help!]

6*) The area of a rectangle with height

x + 1

units and base

x + 4

units is

x^{2} + 5 x + 4

square units.

Which of the following are factors of $x^{2} + 5 x + 4$ ‍?

[I need help!]

Why are we interested in factoring polynomials?

Just as factoring integers turned out to be very useful for a variety of applications, so is polynomial factorization!

Specifically, polynomial factorization is very useful in solving quadratic equations and simplifying rational expressions.

If you'd like to see this, check out the following articles:

What's next?

The next step in the factoring process involves learning how to factor monomials. You can learn about this in our next article.

Want to join the conversation?

Sort by:

msaboosalih
Posted 8 years ago. Direct link to msaboosalih's post “what is a factor”
what is a factor
Button navigates to signup pageComment on msaboosalih's post “what is a factor”
(11 votes)
Answer
- pte-bpotts
  Posted 6 years ago. Direct link to pte-bpotts's post “A number that multiplies ...”
  A number that multiplies to another number.
  Comment on pte-bpotts's post “A number that multiplies ...”
  (36 votes)
NivedhaNK
Posted 4 years ago. Direct link to NivedhaNK's post “The lesson was a little h...”
The lesson was a little hard man...
Button navigates to signup pageComment on NivedhaNK's post “The lesson was a little h...”
(21 votes)
Answer
- mohammad5406
  Posted 4 years ago. Direct link to mohammad5406's post “I'm glad I'm not the only...”
  I'm glad I'm not the only one :/ but if we practice a lot we'll get it eventually!
  Comment on mohammad5406's post “I'm glad I'm not the only...”
  (12 votes)
Garrett Smith
Posted 5 years ago. Direct link to Garrett Smith's post “Hey all. Confused on last...”
Hey all. Confused on last question. Wouldn’t x2+5x be a factor since you just cancel the x2+5x on top and bottom? Or is it not just because of the problem being about Area?
Button navigates to signup pageComment on Garrett Smith's post “Hey all. Confused on last...”
(7 votes)
Answer
- Jonathan Copeland
  Posted 4 years ago. Direct link to Jonathan Copeland's post “I realize this was posted...”
  I realize this was posted 8 months ago, but this is a common mistake so I would like to address it. x^2+5x is not a factor of this expression because it is being added to 4. If that sum were multiplied by 4 instead of added to it, then it would be a factor. The fact that the expression is a sum of x^2+5 and 4 and not a product of the two means that x^2+5 cannot be a factor of x^2+5x+4. I hope that makes sense and clears this up for anyone else wondering the same thing.
  Comment on Jonathan Copeland's post “I realize this was posted...”
  (31 votes)
Dailenys Delgado
Posted 4 years ago. Direct link to Dailenys Delgado's post “This lesson was easy for ...”
This lesson was easy for me how about y`all?
Button navigates to signup pageComment on Dailenys Delgado's post “This lesson was easy for ...”
(7 votes)
Answer
Luis Eduardo
Posted 7 months ago. Direct link to Luis Eduardo's post “What is monomials?”
What is monomials?
Button navigates to signup pageComment on Luis Eduardo's post “What is monomials?”
(4 votes)
Answer
- ellery
  Posted 6 months ago. Direct link to ellery's post “A monomial is a math stat...”
  A monomial is a math statement with exactly one term. A term can be anything with multiplication or division and it can even contain variables, but it cannot contain addition or subtraction. Here are some examples of what is and isn't a monomial.
  
  IS a monomial
  - 9x
  - 14a/2
  - 4
  - 17ab^2
  
  IS NOT a monomial
  - 14x + 4
  - 919 - 7
  - x^2 + 4x + 9
  
  An easy way to remember what a monomial is can be to break it down into its Latin/Greek roots.
  
  Mono - nomial
  mono means one
  nomial means number or term
  
  Hope this helps!
  Comment on ellery's post “A monomial is a math stat...”
  (9 votes)
Kaysie Polydor
Posted 4 years ago. Direct link to Kaysie Polydor's post “What's the easiest way to...”
What's the easiest way to tell a number is a factor of another?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Rebecca
  Posted 4 years ago. Direct link to Rebecca's post “Think about what 2 number...”
  Think about what 2 numbers multiply together to make that number. For example 3 x 4 = 12 therefore 3 and 4 are factors.
  Comment on Rebecca's post “Think about what 2 number...”
  (5 votes)
jose.rodriguez
Posted 4 years ago. Direct link to jose.rodriguez's post “i really don't understand...”
i really don't understand this topic how do you do this
Button navigates to signup pageComment on jose.rodriguez's post “i really don't understand...”
(6 votes)
Answer
WhoDon'tLikeCows/Katherine.Meyer
Posted 5 months ago. Direct link to WhoDon'tLikeCows/Katherine.Meyer's post “How does this work? I mea...”
How does this work?
I mean I get it but it's really hard.

(btw im doing 8th grade math and im in 7th grade.)
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- joshua
  Posted 5 months ago. Direct link to joshua's post “(Assuming the denominator...”
  (Assuming the denominator is a monomial)
  First when you are dividing a polynomial, it's better to take each element separated by + or - in the numerator.
  For example,
  (2x^2 + 6x)/2x
  -> 2x^2/2x + 6x/2x
  This is basically separating a fraction into smaller fractions.
  
  Then the next thing you will do is think as each variable / constant that is being multiplied not placed together.
  Consider 2x^2/2x
  So you can first take 2/2 and compute it, which results to 1.
  Then you consider x^2/x, which is x^(2-1) = x.
  Consider 15x^2y^6 / 10x^4y^3.
  First take 15 / 10, which is 3 / 2. Then consider x^2 / x^4, which is 1 / x^2. Last y^6 / y^3 which is y^3.
  
  The reason why this works is because of the fundamental of algebra.
  Hope this helps
  Button navigates to signup page
  (3 votes)
NibrajahN
Posted 7 months ago. Direct link to NibrajahN's post “the last one got me confu...”
the last one got me confused
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
Benjamin Hao
Posted 3 years ago. Direct link to Benjamin Hao's post “Does the factor of a poly...”
Does the factor of a polynomial always have to be a monomial?

e.g. 3(3x^2 + 3x + 3)
factors would be 3 and (3x^2 + 3x + 3)?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Tahsin Mohammad Rakin
  Posted 3 years ago. Direct link to Tahsin Mohammad Rakin's post “A polynomial can have a m...”
  A polynomial can have a monorail factor if it's terms have a common factor.But it can also have binomial and trionomial factors that can't be factored further.For example: x^3-1=(x-1)(x^2+x+1)
  3(3x^2+3x+3)=9(x^2+x+1) So, the factors are 9 and (x^2+x+1).
  Comment on Tahsin Mohammad Rakin's post “A polynomial can have a m...”
  (2 votes)

Algebra 1

Course: Algebra 1 > Unit 13

What we need to know for this lesson

What we will learn in this lesson

Factors and divisibility in integers

Factors and divisibility in polynomials

Check your understanding

Determining factors and divisibility

Example 1: Is 24x4‍ divisible by 8x3‍ ?

Example 2: Is 4x6‍ a factor of 32x3‍ ?

A summary

Check your understanding

Challenge problems

Why are we interested in factoring polynomials?

What's next?

Want to join the conversation?

Example 1: Is $24 x^{4}$ ‍ divisible by $8 x^{3}$ ‍?

Example 2: Is $4 x^{6}$ ‍ a factor of $32 x^{3}$ ‍?