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Course: Algebra (all content) > Unit 10
Lesson 10: Introduction to factorizationIntro 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 , like . A polynomial is an expression that consists of a sum of monomials, like .
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 , we know that and are factors of .
One number is divisible by another number if the result of the division is an integer.
For example, since and , then is divisible by and . However, since , then is not divisible by .
Notice the mutual relationship between factors and divisibility:
Since (which means is a factor of ), we know that (which means is divisible by ).
In the other direction, since (which means is divisible by ), we know that (which means is a factor of ).
This is true in general: If is a factor of , then is divisible by , 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 .
This means that and are factors of .
Also, one polynomial is divisible by another polynomial if the quotient is also a polynomial.
For example, since and since , then is divisible by and . However, since , we know that is not divisible by .
With polynomials, we can note the same relationship between factors and divisibility as with integers.
In general, if for polynomials , , and , then we know the following:
and are factors of . is divisible by and .
Check your understanding
Determining factors and divisibility
Example 1: Is divisible by ?
To answer this question, we can find and simplify . If the result is a monomial, then is divisible by . If the result is not a monomial, then is not divisible by .
Since the result is a monomial, we know that is divisible by . (This also implies that is a factor of .)
Example 2: Is a factor of ?
If is a factor of , then is divisible by . So let's find and simplify .
Notice that the term is not a monomial since it is a quotient, not a product. Therefore we can conclude that is not a factor of .
A summary
In general, to determine whether one polynomial is divisible by another polynomial , or equivalently whether is a factor of , we can find and examine .
If the simplified form is a polynomial, then is divisible by and is a factor of .
Check your understanding
Challenge problems
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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