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The greatest common factor of two or more terms or values is GCF. The GCF must be a factor of both terms, which means that they must be divisible by the GCF. Within an algebraic expression, a monomial is a single phrase. It can be a number, a variable, or a product of a number and variables, and all variable exponents must be whole numbers (0, 1, 2, 3, ...). The procedure is similar when asked to find the greatest common factor of two or more monomials. Write down the complete factorization of each monomial and then look for the common factors. The Greatest Common Factor will be the sum of all common factors. In order to determine the greatest common factor of two monomials, first determine the prime factorization of each monomial, including all variables (and a – 1 factor if needed). Then compute the sum of all common factors.

For instance, find the greatest common monomial factor of -27p^{2}qr^{5} and 15p^{3}r^{3}. Find the prime factorization of each monomial first.

-27p^{2}qr^{5 }= −1⋅3⋅3⋅3⋅p⋅p⋅q⋅r⋅r⋅r⋅r⋅r

15p^{3}r^{3} = 3⋅5⋅p⋅p⋅p⋅r⋅r⋅r

The most common factors are highlighted in red. Their item is:

3⋅p⋅p⋅r⋅r⋅r

So, the GCF is 3p^{2}r^{3}.

The greatest common monomial factor is the largest term in a given list of terms. Write the prime factorization of each term to find the greatest common monomial factor, and the product of the common factors is the greatest common monomial factor. The procedure is similar when asked to find the greatest common factor of two or more monomials. Write down the complete factorization of each monomial and then look for the common factors. The Greatest Common Factor will be the sum of all common factors. In order to determine the greatest common factor of two monomials, first determine the prime factorization of each monomial, including all variables (and a – 1 factor if needed). Then compute the sum of all common factors.

For instance, find the greatest common monomial factor of -27p^{2}qr^{5} and 15p^{3}r^{3}. Find the prime factorization of each monomial first.

-27p^{2}qr^{5 }= −1⋅3⋅3⋅3⋅p⋅p⋅q⋅r⋅r⋅r⋅r⋅r

15p^{3}r^{3} = 3⋅5⋅p⋅p⋅p⋅r⋅r⋅r

The most common factors are highlighted in red. Their item is:

3⋅p⋅p⋅r⋅r⋅r

So, the GCF is 3p^{2}r^{3}.

Identifying the Greatest Common Factor (GCF): To calculate the GCF of two expressions:

- Divide each coefficient by the number of primes. All variables with exponents should be written in expanded form.
- List all factors in a column, matching common factors at the top. Circle the common factors in each column.
- Reduce the factors that all expressions have in common.
- Multiply the variables as in (Figure).

There are many methods for determining the greatest common factor of numbers. The most efficient method someone uses is determined by the number of numbers they have, their size, and what they intend to do with the results.

Example: Find the GCF of 18 and 27.

Solution:

The factors of 18 are 1, 2, 3, 6, 9, 18.

The factors of 27 are 1, 3, 9, 27.

The common factors of 18 and 27 are 1, 3 and 9.

The greatest common factor of 18 and 27 is 9.

The greatest common factor of two or more terms or values is GCF. The GCF must be a factor of both terms, which means that they must be divisible by the GCF. Within an algebraic expression, a monomial is a single phrase. It can be a number, a variable, or a product of a number and variables, and all variable exponents must be whole numbers (0, 1, 2, 3, ...). The procedure is similar when asked to find the greatest common factor of two or more monomials. Write down the complete factorization of each monomial and then look for the common factors. The Greatest Common Factor will be the sum of all common factors. In order to determine the greatest common factor of two monomials, first determine the prime factorization of each monomial, including all variables (and a – 1 factor if needed). Then compute the sum of all common factors.

For instance, find the greatest common monomial factor of -27p^{2}qr^{5} and 15p^{3}r^{3}. Find the prime factorization of each monomial first.

-27p^{2}qr^{5 }= −1⋅3⋅3⋅3⋅p⋅p⋅q⋅r⋅r⋅r⋅r⋅r

15p^{3}r^{3} = 3⋅5⋅p⋅p⋅p⋅r⋅r⋅r

The most common factors are highlighted in red. Their item is:

3⋅p⋅p⋅r⋅r⋅r

So, the GCF is 3p^{2}r^{3}.

Discovering the greatest common factor for the constants, the greatest common factor for the variable terms, and then using the distributive property to factor out the greatest common factor is the procedure of factoring a monomial from a polynomial (GCF). The first stage in factoring a monomial from a polynomial is to calculate the GCF of the constants and terms. Assume that the polynomial is 3x^{2} + 6x + 9. The GCF of the constants is 3 because 3 * 1 equals 3, 3 * 2 equals 6, and 3 * 3 equals 9. Because 9 does not have a variable associated with it, there is no x term in common. The second stage uses the distributive property to factor out the GCF from each term of the polynomial, resulting in 3(x^{2} + 2x + 3). For instance, if the polynomial is 9x^{3} + 27x^{2} + 24x, the constants factor as 3 * 3, 3 * 9, and 3 * 8, respectively, resulting in a GCF of 3. Each term of the polynomial also has a common variable. As a result, the GCF is 3x, and the answer is 3x (3x^{2} + 9x + 8).

No matter how many terms are in a polynomial, someone should always look for the greatest common factor (GCF) first. If the polynomial has a GCF, factoring the remaining terms is much easier because the remaining terms are less cumbersome once the GCF is factored out.

- Find the GCF of all the polynomial terms.
- Using the GCF, rewrite each term as a product.
- In order to factor the expression, use the Distributive Property 'in reverse.'
- Multiply the factors to see if they are correct.

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