Factoring out a greatest common factor (GCF) is the process of breaking down a polynomial into two factors, a monomial and a polynomial.
Factoring out a GCF makes it possible to remove a factor which is common to all of a polynomial’s terms. It is based on the distributive property of multiplication over addition and subtraction.
Before describing factoring using a GCF, it is important to understand the concept of the GCF. The search for the GCF constitutes the preliminary stage of factoring out a GCF.
To find the GCF, we must:
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Find the GCF of the coefficients for each of the terms and the constant term (if applicable).
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Target the variables that are common to each term.
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Identify the smallest exponent for each targeted variable.
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Identify the GCF of the expression as the product of the GCF of the coefficients and the common variables raised to their smallest exponent values.
For example, the polynomial |30x^6y^3z + 15x^4y^4z^4+20xy^2.|
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Find the GCF of the coefficients for each of the terms
The GCF of |30,| |15,| and |20| is: ||GCF(15,20,30)=5|| -
Target the variables that are common to each term
The variables |x| and |y| are the only variables present in the expression which are common to the three terms. -
Identify the smallest exponent for each targeted variable
The smallest exponent of |x| in the expression is |1.| ||x^1=x|| The smallest exponent of |y| in the expression is |2.| -
Identify the GCF of the expression as the product of the GCF of the coefficients and the common variables raised to their smallest exponent values.
Therefore, the GCF of the polynomial is |5xy^2.|
Factoring out a GCF requires following these steps:
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Find GCF of the coefficients for each term and the constant term (if applicable).
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Identify the smallest exponent for each common variable.
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Determine the first factor (GCF).
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Determine the second factor by dividing all the polynomial’s terms by the GCF.
Factoring out a GCF is always the first operation to perform when factoring a polynomial.
For example, the polynomial |10x^2 + 15x.|
- Find the GCF of the coefficients for each term.
The GCF of |10| and |15| is |5:| ||GCF(10,15)=5||
- Identify the smallest exponent for each common variable.
The variable |x| is the only variable present in the expression and it is common to the two terms. The smallest exponent of |x| in the expression is |1.|
||x^1=x||
- Determine the first factor.
Thus, the GCF of the polynomial is |5x.|
- Determine the second factor by dividing all the polynomial’s terms by GCF.
||\begin{align}10x^2+15x&=\left(\text{GCF}\right) \left(\text{Second factor}\right)\\ &=5x \left(\text{Second factor}\right)\\ &=5x \left(\frac{10x^2+15x}{5x}\right)\\ &=5x\ (2x+3) \end{align}|| The result is: ||5x (2x+3)||
For example, the polynomial |8x^3 + 4x^{2}y + 16x^2.|
- Find the GCF of the coefficients for each term.
The GCF of |8,| |4,| and |16| is |4:| ||GCF(8,4,16)=4||
The variable |x| is the only one common to all the polynomial’s terms. The smallest exponent of |x| in the expression is |2.|
- Identify the smallest exponent for each common variable.
The variable |x| is the only one common to all the polynomial’s terms. The smallest exponent of |x| in the expression is |2.|
- Determine the first factor.
Thus, the GCF of the polynomial is |4x^2.|
- Determine the second factor by dividing all the polynomial’s terms by the GCF. ||\begin{align}8x^3+4x^2y+16x^2&=\left(\text{GCF}\right) \left(\text{Second factor}\right)\\ &=4x^2 \left(\text{Second factor}\right)\\ &=4x^2 \left(\frac{8x^3+4x^2y+16x^2}{4x^2}\right)\\ &=4x^2\ (2x+y+4) \end{align}||
The result is: ||4x^2(2x+y+4)||
Factors obtained during the factorization can be expanded to check whether they are equivalent with the starting polynomial.
Validating the first example:
||5x(2x+3)\overset{?}{=} 10x^2+15x|| Distribute: ||\begin{align}5x(2x+3)&\overset{?}{=} 10x^2+15x\\ (5x\times 2x)+(5x\times 3)&\overset{?}{=} 10x^2+15x\\ 10x^2+15x&=10x^2+15x\end{align}|| The polynomial obtained is indeed equivalent to the starting polynomial, confirming that the factorization is correct.