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.
Find the GCF of 12x^{2} and 20x^{4}
Now, not only we need to expand 12 and 20 into their prime factorizations, we also need to do the same for x^{2} and x^{4} as demonstrated in the figure above.
As you can see from the figure above, the GCF of 12x^{2} and 20x^{4} is 2 × 2 × x × x = 4x^{2}
Example #1
Find the GCF of 15x^{4}, 25x^{3}, and 75x^{5}
First take a look at the figure below and study it carefully.
Thus, the GCF for 15x^{4}, 25x^{3}, and 75x^{5} is 5 × x × x × x = 5x^{3}
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 x^{4}, x^{3}, and x^{5 }is x^{3} since x^{3} is the expression with the smallest exponent. By the same token, the greatest common factor of 3^{2}, 3^{4}, and 3^{3} is 3^{2} since 3^{2} is the expression with the smallest exponent.
Example #2
Find the GCF of w^{3} + w^{2} + w
We can break down this trinomial into three separate expressions: w^{3}, w^{2}, and w.
Since w is the expression with the smallest exponent, w is the greatest common factor.
Example #3
Find the GCF of 4w^{5} + 8w^{4} + 16w^{3}
4w^{5} + 8w^{4} + 16w^{3} = 2^{2}w^{5} + 2^{3}w^{4} + 2^{4}w^{3}
We can break down this trinomial into three separate expressions: 2^{2}w^{5},2^{3}w^{4},and 2^{4}w^{3}
For 2^{2}, 2^{3}, and 2^{4}, the GCF is 2^{2}
For w^{5}, w^{4}, and w^{3}, the GCF is w^{3}
Therefore, the GCF of 4w^{5} + 8w^{4} + 16w^{3} is 2^{2}w^{3} or 4w^{3}
Jan 26, 23 11:44 AM
Jan 25, 23 05:54 AM