In order to find the highest common algebraic factor we can use the following steps.
- First, find the numerical GCF between the coefficients of the terms we are comparing. This GCF will be the coefficient of our highest common algebraic factor.
- Now, check if there any variables (or variables) that appear in both terms. If so, what is the lowest power of each variable that is present? The product of each variable raised to its lowest power will be the algebraic component of our greatest common factor.
- Finally, multiply the numerical and algebraic components to form our highest common algebraic factor. Note that if the numerical GCF was $1$1, we can omit this from our answer.
Let's look at an example.
Worked example
What is the highest common algebraic factor of $2x^2y$2x2y and $6xz$6xz?
Think: We need to look at the numeric coefficient first, then the algebraic variables, and then combine our findings into a solution.
Do: The numerical coefficients are $2$2 and $6$6, so the numerical GCF is $2$2.
The algebraic terms present in the first term are $x$x and $y$y, and those present in the second term are $x$x and $z$z. So $x$x is the only common variable between the two.
The lowest power of $x$x present in either term is $x^1=x$x1=x, so this will be the algebraic component of our highest common algebraic factor.
Solve: Our highest common algebraic factor is $2x$2x.
The same process applies if we are asked to compare $3$3 or more terms, we just need to ensure any common numeric or algebraic variables are common to all terms.
Practice questions
Question 1
What is the highest common algebraic factor of the terms $9ax$9ax and $8ba$8ba?
Question 2
What is the highest common algebraic factor of the terms $4bx$4bx and $6by$6by?
Question 3
What is the highest common algebraic factor of the terms $3a^2mn$3a2mn, $2ya^4x$2ya4x and $5xma^3$5xma3?