Online Common Factoring Calculator (Full Solutions)
Common factoring just got a whole lot easier! This common factoring calculator is designed to help you quickly find the greatest common divisor (GCD) of your algebraic expressions so that you can write the expression in fully factored form.
Whether you’re a student tackling more complex math problems or looking to brush up on your algebra knowledge, this online calculator provides full and detailed solutions for any expression you can think of.
Use the input box below to enter your original expression and see it fully factored with step-by-step solutions!
Common Factoring Calculator
Enter any algebraic expression:
ex: 12b^4c^5 + 4b^3c^6, -b^2c + b^2
Practice your common factoring skills by completing the following exercises. Then, use the calculator to check your work!
- \(-30p^5 + 45p^3 – 15p^2\)
- \(15x^2 + 25x\)
- \(-12a^3b + 18a^2b^2\)
- \(8m^4n^3 – 20m^2n^5\)
- \(21x^3y^2 – 14x^2y^3\)
How to Common Factor an Expression
When common factoring an expression, we consider two things:
- the greatest common factor of the coefficients
- the greatest common factor of the variables
As a first step, I usually suggest looking for common factors amongst the coefficients of the expression. Look at the numbers in front of the variables and start listing the factors of each number.
From this set of numbers, the largest number that divides all terms evenly is called the greatest common divisor (GCD) or highest common factor (HCF).
I normally recommend looking for positive factors first, but just note that your GCF can also be a negative number (more on that later!).
The second step is to look for the GCF of the variable portion of each term. To do this, we look to the exponent of each variable. The GCF will be the lowest exponent on the variable that is common amongst all terms.
Lastly, we divide each term of the expression by the entire GCF. This includes both the coefficient and the variable portions.
Let’s take a look at an example!
Example 1: Common Factoring
Original expression: \(12x+18x^2\)
Let’s start by finding the GCF of the coefficients. To do this, we write out the list of factors of each coefficient.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
Next, we can identify the greatest common divisor amongst each set of numbers.
- Common divisors: 1, 2, 3, 6
- Greatest common factor (GCF): 6
The next step is to look at the variable portion of each term. Notice how the lowest exponent is 1.
Our terms can be thought of as:
- \(12x=12(x)\)
- \(18x^2=18(x)(x)\)
Notice how the largest number of x’s that is common between each term is 1x. Therefore, x will be our GCF for our variable portion.
So overall, our GCF is \(6x\).
The last step is to divide each term of our expression by our GCF. Doing this results in:
\[\frac{12x}{6x}=2\]
\[\frac{18x^2}{6x}=3x\]
We then write the result in brackets with the common factor placed on the outside:
\[6x(2+3x)\]
Therefore, we say that our original expression of \(12x+18x^2\) factors into \(6x(2+3x)\).
The GCF calculator above handles all these steps automatically, making it a reliable math solver for factoring!
Special Cases in Common Factoring
When factoring, sometimes you’ll encounter a negative number as a leading term. In such cases, it’s usually best to factor out a negative factor to keep the expression clean and simple.
Let’s look at another example!
Example 2: Common Factoring
Original polynomial: \(-8x^3 + 12x^2\)
- Factors of -8: -1, -2, -4, -8 (and their positive counterparts)
- Factors of 12: 1, 2, 3, 4, 6, 12
- Common factors: 1, 2, 4
- Greatest common factor (GCF): 4 (and since leading coefficient is negative, we factor out -4)
Next, we consider the variable portion of each term. In this case, we can write our terms as:
- \(-8x^3=-8(x)(x)(x)\)
- \(12x^2=12(x)(x)\)
The greatest number of x’s common to each term is 2. Therefore, \(x^2\) will be the GCF for the variable portion. This means our overall GCF will be \(-4x^2\).
Dividing each term of the original expression by the GCF results in:
\[\frac{-8x^3}{-4x^2}=2x\]
\[\frac{12x^2}{-4x^2}=-3\]
Therefore, our original expression of \(-8x^3 + 12x^2\) factors into \(-4x^2(2x-3)\).
The factoring calculator above will quickly recognize these special cases and provide the correct factored form for you!
Using This Common Factoring Calculator
This online greatest common factor calculator is a powerful tool for any students looking to improve their factoring skills and tackle even the most challenging problems.
Whether you’re working with positive integers, searching through factor pairs, or listing the factors of a number, this calculator will walk you through every step to successfully common factor polynomials.
When factoring, it can be tough to identify the factors when given numbers that are large or variables with large exponents. This tool helps you remove some of the guess work, while providing you with an opportunity to strengthen your understanding using the full solutions provided.
Factoring is a very important skill that you will apply in a wide range of mathematical problems. Use this calculator to help complete your practice problems so that you can master the skill of common factoring!
Looking for more helpful online math calculators like this one? Check out my trinomial factoring calculator and my difference of squares factoring calculator!
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