Finding the greatest common factor of an expression

In greatest common factor, finding the greatest common factor of numbers was the goal. In this lesson, we will do the same thing, but we will take the concept to the next level by looking for the greatest common factor of expressions.

Recall that if we are looking for the GCF of 12 and 20, we need to first factor each number into its prime factorization as shown in the figure below.

As you can see, 2 × 2 = 4 is the greatest factor 12 and 20 have in common, so 4 is the GCF of 12 and 20.

Greatest common factor

Find the GCF of 12x2 and 20x4

Now, not only we need to expand 12 and 20 into their prime factorizations, we also need to do the same for x2 and x4 as demonstrated in the figure above.

As you can see from the figure above, the GCF of 12x2 and 20x4 is 2 × 2 × x × x = 4x2

A few more examples showing how finding the greatest common factor of expressions work


Example #1

Find the GCF of 15x4, 25x3, and 75x5

First take a look at the figure below and study it carefully.

common factor of  15x^4, 25x^3, and 75x^5

Thus, the GCF for 15x4, 25x3, and 75x5 is 5 × x × x × x = 5x3

There is a faster way to find the greatest common factor of an expression.

When there are many exponential expressions that have the same variable or same number as base, the greatest common factor is the expression with the smallest exponent.

For example, the greatest common factor of x4, x3, and xis x3 since x3 is the expression with the smallest exponent. By the same token, the greatest common factor of 32, 34, and 33 is 32 since 32 is the expression with the smallest exponent.

Example #2

Find the GCF of w3 + w2 + w 

We can break down this trinomial into three separate expressions: w3, w2, and w.

Since w is the expression with the smallest exponent, w is the greatest common factor.

Example #3

Find the GCF of 4w5 + 8w4 + 16w3

4w5 + 8w4 + 16w3 = 22w5 + 23w4 + 24w3

We can break down this trinomial into three separate expressions: 22w5,23w4,and 24w3

For 22, 23, and 24, the GCF is 22

For w5, w4, and w3, the GCF is w3

Therefore, the GCF of 4w5 + 8w4 + 16w3 is 22w3 or 4w3

Finding the greatest common factor quiz. See how well you understand this lesson.

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