Lesson Explainer: Factoring Using the Highest Common Factor

In this explainer, we will learn how to factor algebraic expressions by identifying the highest common factor (HCF).

The distributive property of multiplication over addition can be used to multiply polynomials. However, just like with real numbers, we can use this property to factor expressions by using this property in the other direction.

To see how to do this, let’s first recall how we can distribute and factor real expressions by using an area model. We can evaluate the expression by considering the areas of two rectangles.

We note that each product is the area of a rectangle with dimensions equal to each factor. We could evaluate each area to get . However, since both rectangles share a dimension, we can combine these into a larger rectangle.

We can then calculate the area of this rectangle to evaluate the expression as follows:

This is a justification of the distributive property of multiplication over addition, and it gives us a useful example of not only distributing over a sum but also factoring over a sum. To see this, note that we showed . We can think of this as taking out the shared factor of 2 from both terms:

We can apply this process to any numbers or variables using the distributive property. In particular, we can extend this to algebraic expressions by noting that variables such as and are also numbers. This means that all of the properties that hold for numbers will also hold for algebraic expressions. So, we can take out shared factors in algebraic expressions.

For example, consider the expression . We can see that both 2 and 4 share a factor of 2. In fact, it is their highest common factor. We can use the distributive property to take out this shared factor of 2 as follows:

We can confirm that this works by distributing 2 over the parentheses on the right-hand side of the equation. However, this is not the highest common factor of both terms. We can also see that both terms have a factor of . We have

Once again, we can verify this by distributing the right-hand side of the equation. It is also worth pointing out that 1 and share no more common factors other than 1, so we cannot factor out anything else.

In the above example, we noted that after taking out a factor of , we could not factor anything other than 1 from the expression. We call the highest common factor of the two terms. In general, we can find the highest common monomial factor of an algebraic expression by finding the highest common factor of the coefficients and then multiplying it by each common variable raised to the lowest exponent in any term.

For example, if we want to find the highest common factor of the monomial terms in the expression , we first write each term as a product of factors:

The numbers 2, 3, and are all factors of the three terms. Hence, the highest common factor is given by their product:

Note that this process is like that of finding the highest common factor of numbers; the only difference is that instead of having only prime factors, we also have each variable as a factor.

We can now then take out this highest shared factor of :

We can generalize this process as follows.

How To: Finding the Highest Common Monomial Factor of a Polynomial

Find the highest common factor of the coefficients.
For each variable, find the term in which this variable is raised to the lowest exponent.
Multiply the highest common factor of the coefficients with each common variable raised to the lowest exponent that appears in the expression.

Let’s now see some examples of using this process to determine the highest common monomial factor of given polynomials.

Example 1: Using a Diagram to Factor an Expression by Finding the Highest Common Factor

Using the diagram, factor .

Answer

To factor this expression using the area model, we need to make the dimensions of the rectangle of area 12 share a dimension with the rectangle of area . We can do this by noting that . We can then combine the two smaller rectangles into a larger rectangle with dimensions 4 and as shown.

The sum of the areas of the two smaller rectangles must be equal to the area of the larger rectangle, so we must have

In our next example, we will completely factor a given polynomial expression.

Example 2: Factoring an Expression by Finding the Highest Common Factor

Factor completely.

Answer

Factoring a polynomial expression completely means that there must be no shared factors remaining in the terms. This is equivalent to taking out the highest common monomial factor.

We can find the highest common monomial factor by first noting that and only appear in one term, so they do not have any common factor. Hence, the highest common factor of and will be the highest common factor of the coefficients. Since the coefficients are both equal to 15, we take out the factor of 15 to get

In our next example, we will determine the greatest common factor of a given two-term polynomial.

Example 3: Finding the Highest Common Factor of a Single Variable Expression

Find the greatest common factor of the two terms in this expression: .

Answer

We first note that all of the terms in this expression are monomials, so its greatest common factor will be the greatest common monomial factor. We then recall that we can find the greatest common monomial factor of an algebraic expression by finding the greatest common factor of the coefficients and then multiplying this value by each common variable raised to the lowest exponent in any term.

Let’s start by finding the greatest common factor of the coefficients. We do this by factoring each coefficient into primes. We have and . Taking the common prime factors, we know that the greatest common factor of the coefficients is 2.

Next, we want to find the greatest factor of in the two terms. Since , we can see that both terms share a factor of , and we cannot take out any further factors of .

Hence, the greatest common factor of the expression is .

In our next example, we will factor a given expression for the surface area of a square prism.

Example 4: Finding the Highest Common Factor to Solve a Geometric Problem

The total surface area of the square prism shown can be expressed as . Write the factorized expression of the surface area of the square prism.

Answer

First, we note that all of the terms in this expression are monomials, so its greatest common factor will be the greatest common monomial factor. We then recall that we can find the greatest common monomial factor of an algebraic expression by finding the greatest common factor of the coefficients and then multiplying this value by each common variable raised to the lowest exponent in any term.

Let’s start with the coefficients. We see that 2 is a divisor of 4, so the highest common factor of the coefficients is 2 itself. Next, we want to find the greatest factor of in both terms. We see that the second term only has a single factor of , so this is the greatest factor of we can take out. Finally, we see that the first term does not have a factor of , so there are no shared factors of .

Hence, the highest common factor of the expression is . We can take this factor out using the distributive property to get

Therefore, the total surface area of the square prism in factored form is .

It is worth noting that the process of factoring does not need to stop at monomials. For example, we can factor by noting that both terms share a factor of . We can take this factor out using the distributive property to get

We can also note that the terms 6x and 2 share a factor of 2. Taking this factor out as well gives

We call the highest common factor of the terms since there are no more common factors between and 1 other than 1 itself.

In our next example, we will find the highest common factor of an algebraic expression by identifying a shared binomial factor.

Example 5: Identifying Binomial Factors of Polynomial Expressions

Factorize fully .

Answer

Since this expression contains binomial factors, to fully factor this expression, we need to first look for shared binomial factors. We can see that both terms have a factor of . Taking this factor out by using the distributive property and simplifying yields

We can then note that and 12 share no common factors other than 1, so we cannot factor further.

Hence,

In our final example, we will find the greatest common factor of a given algebraic expression.

Example 6: Identifying Binomial Factors of Polynomial Expressions

By identifying the GCF of the terms, factorize the expression .

Answer

Since this expression contains binomial factors, to find the greatest common factor of the expression, we first need to check if there are any shared binomial factors. At first, it may seem like there are no shared binomial factors; however, we can recall that since .

This motivates taking a factor of out of the second term to get

We then see that both terms share a factor of . We can take this factor out using the distributive property to get

We then need to check if and share any common factors. We note that 7 is prime and has no common factors with 6 and that both terms have a different variable, so they share no common factors other than 1.

Hence, the expression is fully factored as

Let’s finish by recapping some of the important points from this explainer.

Key Points

We can find the highest common monomial factor of a polynomial by using the following steps:
1. Find the highest common factor of the coefficients.
2. For each common variable, find the term in which this variable is raised to the lowest exponent.
3. Multiply the highest common factor of the coefficients with each common variable raised to the lowest exponent that appears in the expression.
The highest common factor of an algebraic expression is the expression that when factored out leaves the remaining terms with no shared factors other than 1.
The highest common factor of an algebraic expression can be a polynomial; it is not restricted to monomials.