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Intro to factors & divisibility

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, squared. A polynomial is an expression that consists of a sum of monomials, like 3, x, squared, plus, 6, x, minus, 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, equals, 2, dot, 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 start fraction, 15, divided by, 3, end fraction, equals, 5 and start fraction, 15, divided by, 5, end fraction, equals, 3, then 15 is divisible by 3 and 5. However, since start fraction, 9, divided by, 4, end fraction, equals, 2, point, 25, then 9 is not divisible by 4.
Notice the mutual relationship between factors and divisibility:
Since start color #e07d10, 14, end color #e07d10, equals, start color #11accd, 2, end color #11accd, dot, 7 (which means 2 is a factor of 14), we know that start fraction, start color #e07d10, 14, end color #e07d10, divided by, start color #11accd, 2, end color #11accd, end fraction, equals, 7 (which means 14 is divisible by 2).
start underbrace, start color #a75a05, 14, end color #a75a05, equals, start color #0c7f99, 2, end color #0c7f99, dot, 7, end underbrace, start subscript, 2, start text, space, i, s, space, a, space, f, a, c, t, o, r, space, o, f, space, end text, 14, end subscript, \longrightarrow, start underbrace, start fraction, start color #a75a05, 14, end color #a75a05, divided by, start color #0c7f99, 2, end color #0c7f99, end fraction, equals, 7, end underbrace, start subscript, 14, start text, space, i, s, space, d, i, v, i, s, i, b, l, e, space, b, y, space, end text, 2, end subscript
In the other direction, since start fraction, start color #e07d10, 15, end color #e07d10, divided by, start color #11accd, 3, end color #11accd, end fraction, equals, 5 (which means 15 is divisible by 3), we know that start color #e07d10, 15, end color #e07d10, equals, start color #11accd, 3, end color #11accd, dot, 5 (which means 3 is a factor of 15).
start underbrace, start fraction, start color #a75a05, 15, end color #a75a05, divided by, start color #0c7f99, 3, end color #0c7f99, end fraction, equals, 5, end underbrace, start subscript, 15, start text, space, i, s, space, d, i, v, i, s, i, b, l, e, space, b, y, space, end text, 3, end subscript, \longrightarrow, start underbrace, start color #a75a05, 15, end color #a75a05, equals, start color #0c7f99, 3, end color #0c7f99, dot, 5, end underbrace, start subscript, 3, start text, space, i, s, space, a, space, f, a, c, t, o, r, space, o, f, space, end text, 15, end subscript
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, left parenthesis, x, plus, 3, right parenthesis, equals, 2, x, squared, plus, 6, x. This means that 2, x and x, plus, 3 are factors of 2, x, squared, plus, 6, x.
Also, one polynomial is divisible by another polynomial if the quotient is also a polynomial.
For example, since start fraction, 6, x, squared, divided by, 3, x, end fraction, equals, 2, x and since start fraction, 6, x, squared, divided by, 2, x, end fraction, equals, 3, x, then 6, x, squared is divisible by 3, x and 2, x. However, since start fraction, 4, x, divided by, 2, x, squared, end fraction, equals, start fraction, 2, divided by, x, end fraction, we know that 4, x is not divisible by 2, x, squared.
With polynomials, we can note the same relationship between factors and divisibility as with integers.
start underbrace, start color #0c7f99, 2, x, end color #0c7f99, dot, left parenthesis, x, plus, 3, right parenthesis, equals, start color #a75a05, 2, x, squared, plus, 6, x, end color #a75a05, end underbrace, start subscript, 2, x, start text, space, i, s, space, a, space, f, a, c, t, o, r, space, o, f, space, end text, 2, x, squared, plus, 6, x, end subscript, \longrightarrow, start underbrace, start fraction, start color #a75a05, 2, x, squared, plus, 6, x, end color #a75a05, divided by, start color #0c7f99, 2, x, end color #0c7f99, end fraction, equals, x, plus, 3, end underbrace, start subscript, 2, x, squared, plus, 6, x, start text, space, i, s, space, d, i, v, i, s, i, b, l, e, space, b, y, space, end text, 2, x, end subscript
start underbrace, start fraction, start color #a75a05, 6, x, squared, end color #a75a05, divided by, start color #0c7f99, 3, x, end color #0c7f99, end fraction, equals, 2, x, end underbrace, start subscript, 6, x, squared, start text, space, i, s, space, d, i, v, i, s, i, b, l, e, space, b, y, space, end text, 3, x, end subscript, \longrightarrow, start underbrace, start color #0c7f99, 3, x, end color #0c7f99, dot, left parenthesis, 2, x, right parenthesis, equals, start color #a75a05, 6, x, squared, end color #a75a05, end underbrace, start subscript, 3, x, start text, space, i, s, space, a, space, f, a, c, t, o, r, space, o, f, space, end text, 6, x, squared, end subscript
In general, if p, equals, q, dot, 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, left parenthesis, x, plus, 2, right parenthesis, equals, 3, x, squared, plus, 6, x.
x, plus, 2 is
3, x, squared, plus, 6, x, and 3, x, squared, plus, 6, x is
x, plus, 2.

2) A teacher writes the following product on the board:
left parenthesis, 3, x, squared, right parenthesis, left parenthesis, 4, x, right parenthesis, equals, 12, x, cubed
Miles concludes that 3, x, squared is a factor of 12, x, cubed.
Jude concludes that 12, x, cubed is divisible by 4, x.
Who is correct?
Choose 1 answer:
Choose 1 answer:

Determining factors and divisibility

Example 1: Is 24, x, start superscript, 4, end superscript divisible by 8, x, cubed?

To answer this question, we can find and simplify start fraction, 24, x, start superscript, 4, end superscript, divided by, 8, x, cubed, end fraction. If the result is a monomial, then 24, x, start superscript, 4, end superscript is divisible by 8, x, cubed. If the result is not a monomial, then 24, x, start superscript, 4, end superscript is not divisible by 8, x, cubed.
24x48x3=248x4x3=3x1aman=amn=3x\begin{aligned}\dfrac{24x^4}{8x^3}&=\dfrac{24}{8}\cdot\dfrac{x^4}{x^3}\\ \\ &=3\cdot x^1&&{\gray{\dfrac{a^m}{a^n}=a^{m-n}}}\\ \\ &=3x \end{aligned}
Since the result is a monomial, we know that 24, x, start superscript, 4, end superscript is divisible by 8, x, cubed. (This also implies that 8, x, cubed is a factor of 24, x, start superscript, 4, end superscript.)

Example 2: Is 4, x, start superscript, 6, end superscript a factor of 32, x, cubed?

If 4, x, start superscript, 6, end superscript is a factor of 32, x, cubed, then 32, x, cubed is divisible by 4, x, start superscript, 6, end superscript. So let's find and simplify start fraction, 32, x, cubed, divided by, 4, x, start superscript, 6, end superscript, end fraction.
32x34x6=324x3x6=8x3aman=amn=81x3am=1am=8x3\begin{aligned}\dfrac{32x^3}{4x^6}&=\dfrac{32}{4}\cdot\dfrac{x^3}{x^6}\\ \\ &=8\cdot x^{-3}&&{\gray{\dfrac{a^m}{a^n}=a^{m-n}}}\\ \\ &=8\cdot \dfrac{1}{x^3}&&{\gray{a^{-m}=\dfrac{1}{a^m}}}\\ \\ &=\dfrac{8}{x^3} \end{aligned}
Notice that the term start fraction, 8, divided by, x, cubed, end fraction is not a monomial since it is a quotient, not a product. Therefore we can conclude that 4, x, start superscript, 6, end superscript is not a factor of 32, x, cubed.

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 start fraction, p, left parenthesis, x, right parenthesis, divided by, q, left parenthesis, x, right parenthesis, end fraction.
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, start superscript, 4, end superscript divisible by 2, x, squared?
Choose 1 answer:
Choose 1 answer:

4) Is 12, x, squared a factor of 6, x?
Choose 1 answer:
Choose 1 answer:

Challenge problems

5*) Which of the following monomials are factors of 15, x, squared, y, start superscript, 6, end superscript ?
 
Factor
Not Factor
3, x, squared, y, start superscript, 5, end superscript
5, x
10, x, start superscript, 4, end superscript, y, cubed

6*) The area of a rectangle with height x, plus, 1 units and base x, plus, 4 units is x, squared, plus, 5, x, plus, 4 square units.
An area model for a rectangle that has a height of x plus one and a width of x plus four. The area of the rectangle is x squared plus five x plus four.
Which of the following are factors of x, squared, plus, 5, x, plus, 4?
Choose all answers that apply:
Choose all answers that apply:

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.

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