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2 Tutorials that teach Finding the Greatest Common Factor of a Polynomial
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Finding the Greatest Common Factor of a Polynomial

Finding the Greatest Common Factor of a Polynomial

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In this lesson, students will learn how to determine the greatest common factor of a polynomial expression, and then use it to factor the expression.

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Tutorial
This tutorial covers how to find the greatest common factor of a polynomial, through the definition and discussion of:
  1. Factors and Terms in Polynomials
  2. Distribution and Factoring
  3. Factorization
  4. Factoring Polynomials


1. Factors and Terms in Polynomials

A polynomial is a mathematical expression involving one or more terms. Terms are separated by addition or subtraction, and each term may contain coefficients, variables, and exponents that are multiplied together. For example, the following polynomial has three terms:

File:1191-fact1.PNG

In the first term, 7 is the coefficient, y is the base, and 4 is the exponent. The coefficients and variables of a term are factors of that term. An expression can also be a factor of a polynomial if it is multiplied by another expression to get the original polynomial as a product.

In this expression, x minus 1 and 2x plus 5 are both factors of the polynomial.

File:1192-fact2.PNG


2. Distribution and Factoring

Distribution involves multiplying a factor outside the parentheses by all the terms inside the parentheses.

As shown in the formula below, the a variable is multiplied by both the b and the c inside the parentheses.
a left parenthesis b plus c right parenthesis equals a b plus a c

Factoring, on the other hand, is the reverse process of distributing, in which a common factor is factored out of two or more terms, and written outside of the parentheses.

In the expression below, the variable a is the common factor of ab and ac. You can factor it out by writing it on the outside of the parentheses, and writing b plus c inside the parentheses.
a b plus a c equals a left parenthesis b plus c right parenthesis

3. Factorization

Being able to factor a polynomial is useful for:

  • Simplifying polynomials
  • Canceling common factors
  • Identifying information, such as intercepts, on graphs

The process of finding common factors of a polynomial works in the same manner as finding common factors of numbers. To “factor” means to break down a number into smaller numbers that, when multiplied together, give you the original number.

The number 30 can be broken down into smaller numbers, as shown below. When multiplied together, these smaller numbers are equal to 30. These smaller numbers are also the prime factorization of 30, because 2, 3, and 5 are all prime numbers.
30 equals 2 cross times 3 cross times 5

Writing the prime factorization of all terms in a polynomial helps identify common factors and the greatest common factor.


4. Factoring Polynomials

Suppose you want to factor the following polynomial. To do this, you need to factor or break down each individual term into its prime components.

File:1193-fact3.PNG

Looking at these three factorizations, you can see that the common factor of each of these terms is 3. If terms have two or more common factors, the factors can be multiplied together to find the greatest common factor.

Here is another example. Again, to factor this polynomial, you need to find the prime factorization of each of the three terms.

File:1194-fact4.PNG

You can see that all of your terms have two common factors. They each have a positive 2, and they each have a variable x. To find your greatest common factor, you multiply these two common factors, 2 and x. Therefore, your greatest common factor is 2x.

Once you have factored out the greatest common factor, the expression inside the parentheses will be the remaining factors of each term. From your first term, your remaining factors are 5x and x or 5x^2. In your second term, the remaining factors are a -2 and x, or -2x. Finally, in your last term, the remaining factor is 2.

2 x left parenthesis 5 x squared minus 2 x plus 2 right parenthesis

You can check to see that you have factored the polynomial correctly by distributing the greatest common factor back into each term inside the parentheses:

table attributes columnalign left end attributes row cell 2 x left parenthesis 5 x squared minus 2 x plus 2 right parenthesis equals end cell row cell 10 x cubed minus 4 x squared plus 4 x end cell end table

Because you have arrived back at your original expression, you have factored your polynomial correctly.

Use what you’ve learned about prime factorization of a polynomial to factor the following expression. Remember, you’ll need to factor each of the three terms.

File:1195-fact5.PNG

Here, again, you see that you have two common factors in each term, as each term has a 2 and a 3. Multiplying these common factors together provides your greatest common factor of 6. Since 6 is your greatest common factor, write it on the outside of the parentheses. Inside the parentheses, write the remaining factors of each term. Note that in your last term, you have already factored out both of the factors of 6, which means that you only have a 1 remaining.
6 left parenthesis 2 x squared minus 3 x plus 1 right parenthesis
Check to see that you have factored your polynomial correctly by distributing the 6 back into each of the three terms in the parentheses.
table attributes columnalign left end attributes row cell 6 left parenthesis 2 x squared minus 3 x plus 1 right parenthesis equals end cell row cell 12 x squared minus 18 x plus 6 end cell end table

Since you have arrived back at your original expression, you have indeed factored your polynomial correctly.
Today you learned about factors and terms in polynomials, which are expressions involving one or more terms. You reviewed the process of distribution, and learned that factoring is the reverse process of distributing, in which a common factor is factored out of two or more terms and written outside of the parentheses. You also learned about factorization, and how finding the prime factorization of all terms in a polynomial helps identify common factors and the greatest common factor. Lastly, you learned how to apply all of this knowledge in factoring polynomials.

Source: This work is adapted from Sophia author Colleen Atakpu.