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Lesson Explainer: Factoring Using the Highest Common Factor Mathematics • 7th Grade

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 (2×4)+(2×3) 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 (2×4)+(2×3)=8+6=14. 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: (2×4)+(2×3)=2×(3+4)=2×7=14.

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 (2×4)+(2×3)=2×(3+4). We can think of this as taking out the shared factor of 2 from both terms: (2×4)+(2×3)=2×(3+4).

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 𝑥 and 𝑦 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 2𝑥+4𝑥𝑦. 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: 2𝑥+4𝑥𝑦=2(𝑥+2𝑥𝑦).

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 𝑥. We have 2𝑥+4𝑥𝑦=2𝑥(1+2𝑦).

Once again, we can verify this by distributing the right-hand side of the equation. It is also worth pointing out that 1 and 2𝑦 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 2𝑥, we could not factor anything other than 1 from the expression. We call 2𝑥 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 12𝑥𝑦+6𝑥𝑦+18𝑥, we first write each term as a product of factors: 12𝑥𝑦=23𝑥𝑦,6𝑥𝑦=23𝑥𝑦,18𝑥=23𝑥.

The numbers 2, 3, and 𝑥 are all factors of the three terms. Hence, the highest common factor is given by their product: 23𝑥=6𝑥.

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 6𝑥: 12𝑥𝑦+6𝑥𝑦+18𝑥=6𝑥×2𝑥𝑦+6𝑥×𝑦+6𝑥×3=6𝑥2𝑥𝑦+𝑦+3.

We can generalize this process as follows.

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

  1. Find the highest common factor of the coefficients.
  2. For each 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.

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 4𝑥+12.

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 4𝑥. We can do this by noting that 12=4×3. We can then combine the two smaller rectangles into a larger rectangle with dimensions 4 and 𝑥+3 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 4𝑥+12=4(𝑥+3).

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

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

Factor 15𝑒+15𝑓 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 𝑒 and 𝑓 only appear in one term, so they do not have any common factor. Hence, the highest common factor of 15𝑒 and 15𝑓 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 15𝑒+15𝑓=15(𝑒+𝑓).

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: 4𝑥18𝑥.

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 4=2 and 18=2×3. 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 𝑥 in the two terms. Since 𝑥=𝑥×𝑥, we can see that both terms share a factor of 𝑥, and we cannot take out any further factors of 𝑥.

Hence, the greatest common factor of the expression is 2𝑥.

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 2𝑥+4𝑥𝑦. 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 𝑥 in both terms. We see that the second term only has a single factor of 𝑥, so this is the greatest factor of 𝑥 we can take out. Finally, we see that the first term does not have a factor of 𝑦, so there are no shared factors of 𝑦.

Hence, the highest common factor of the expression is 2𝑥. We can take this factor out using the distributive property to get 2𝑥+4𝑥𝑦=(2𝑥×𝑥)+(2𝑥×2𝑦)=2𝑥(𝑥+2𝑦).

Therefore, the total surface area of the square prism in factored form is 2𝑥(𝑥+2𝑦).

It is worth noting that the process of factoring does not need to stop at monomials. For example, we can factor 6𝑥(𝑥+1)+2(𝑥+1) by noting that both terms share a factor of (𝑥+1). We can take this factor out using the distributive property to get (𝑥+1)×(6𝑥+2).

We can also note that the terms 6x and 2 share a factor of 2. Taking this factor out as well gives 6𝑥(𝑥+1)+2(𝑥+1)=2(𝑥+1)(3𝑥+1).

We call 2(𝑥+1) the highest common factor of the terms since there are no more common factors between 3𝑥 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 (𝑎10)(𝑎+8)2(𝑎+8).

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 (𝑎+8). Taking this factor out by using the distributive property and simplifying yields (𝑎10)(𝑎+8)2(𝑎+8)=(𝑎+8)(𝑎102)=(𝑎+8)(𝑎12).

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

Hence, (𝑎10)(𝑎+8)2(𝑎+8)=(𝑎+8)(𝑎12).

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 7𝑛(7𝑥8𝑦)+6𝑚(8𝑦7𝑥).

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 𝑎𝑏=(𝑏𝑎) since 53=(35).

This motivates taking a factor of 1 out of the second term to get 7𝑛(7𝑥8𝑦)+6𝑚(8𝑦7𝑥)=7𝑛(7𝑥8𝑦)6𝑚(7𝑥8𝑦).

We then see that both terms share a factor of 7𝑥8𝑦. We can take this factor out using the distributive property to get 7𝑛(7𝑥8𝑦)6𝑚(7𝑥8𝑦)=(7𝑥8𝑦)(7𝑛6𝑚).

We then need to check if 7𝑛 and 6𝑚 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 7𝑛(7𝑥8𝑦)+6𝑚(8𝑦7𝑥)=(7𝑛6𝑚)(7𝑥8𝑦).

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.

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