When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex] because it is the largest number that divides evenly into both [latex]16[/latex] and [latex]20[/latex] The GCF of polynomials works the same way: [latex]4x[/latex] is the GCF of [latex]16x[/latex] and [latex]20x^2[/latex] because it is the largest polynomial that divides evenly into both [latex]16x[/latex] and [latex]20x^2[/latex].
When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.
Greatest Common Factor
The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.
Given a polynomial expression, factor out the greatest common factor.
- Identify the GCF of the coefficients.
- Identify the GCF of the variables.
- Combine to find the GCF of the expression.
- Determine what the GCF needs to be multiplied by to obtain each term in the expression.
- Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.
- Factor [latex]6x^3y^3+45x^2y^2+21xy[/latex].
Analysis
After factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\left(3xy\right)\left(2x^2y^2+15xy+7\right)=6x^3y^3+45x^2y^2+21xy[/latex].
- Factor [latex]x\left(b^2-a\right)+6\left(b^2-a\right)[/latex] by pulling out the GCF.
Solution
Access for free at //openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
In maths, the GCF (greatest common factor) of two or more monomials is the product of all their common prime factors. For example, the GCF of 6x and 4x^2 is 2x.
To understand how to factor out common factors, we must understand the distributive property. The formula for distributive property is a(b+c)=ab+ac.
Factoring out the greatest common factor (GCF) of a Polynomial
To factor the GCF out of a polynomial, we have to follow the below-mentioned steps carefully without any fail:
- Determine the GCF of all the terms in the polynomial.
- Express each term as a product of the GCF and another factor.
- Use the distributive property to factor out the GCF.
Try to solve the given polynomial GCF Factoring by using these simple steps manually or else make use of the onlinecalculator.guru provided Factor out the GCF from the Polynomial Calculator to get the result in split second.
1. What is Factoring?
Factoring is a process of modifying expression from a sum or difference of terms to a product of factors.
2. How do you factor out the GCF from Polynomials?
Factoring polynomials by using the Greatest common factor is simple & easy to perform using the steps provided on our page. So, follow them from the above sections & factor out the GCF of Polynomials.
3. How to solve factoring polynomials by taking the greatest common factor using Factor out the GCF from the Polynomial Calculator?
By using Factor out the GCF from the Polynomial Calculator, solving the factoring polynomials by a common factor is very easy & faster. You just need to give the input polynomials in the input field & press the calculate button.
No matter how many terms a polynomial has, you always want to check for a greatest common factor (GCF) first. If the polynomial has a GCF, factoring the rest of the polynomial is much easier because once you factor out the GCF, the remaining terms will be less cumbersome. If the GCF includes a variable, your job
becomes even easier. When solving for x in a polynomial equation, if you forget to factor out the GCF, you may miss a solution, and that could mix you up in more ways than one! Without that solution, you could end up with an incorrect graph for your polynomial. And then all your work would be for nothing! To factor the polynomial 6x4 – 12x3 + 4x2, for example, follow these steps: Break down every term into
prime factors. This step expands the original expression to
Look for factors that appear in every single term to determine the GCF.
In this example, you can see one 2 and two x’s in every term:
The GCF here is 2x2.
Factor the GCF out from every term in front of parentheses and group the remnants inside the parentheses.
You now have
Multiply each term to simplify.
The simplified form of the expression you find in Step 3 is 2x2(3x2 – 6x + 2).
To see if you factored correctly, distribute the GCF and see if you obtain your original polynomial. If you multiply the 2x2 inside the parentheses, you get 6x4 – 12x3 + 4x2. You can now say with confidence that 2x2 is the GCF.
About This Article
This article can be found in the category:
- Pre-Calculus ,