Chapter A.3, Problem 144AYU

To determine

To find: Explain to a fellow student what you look for first when presented with a factoring problem. What do you do next?

Answer to Problem 144AYU

Here are some guidelines that you can use to help you factor polynomials. I have listed all of the guidelines first and then I will have some examples of each type. To view examples of each type of factoring problem, click on the links below.

1. Factor out anything that all the terms have in common. This is called the greatest common factor or GCF. This step can either be done at the beginning or the end of the problem. I suggest that you do this step first; it will make the numbers smaller and easier to use.

2. Count the number of terms. Depending on how many terms the problem has, you will use a different factoring technique.

a. Four Terms – If the problem has four terms, you will use a technique called factoring by grouping.

b. Three Terms – If the problem has three terms, you will use a technique that also requires factoring by grouping, but there are a few other steps required to end up with four terms. These steps are shown in detail in the examples. There are two types of examples:

Factoring when the leading coefficient is 1.

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power.

Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together.

Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to $x$.

Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to $x$ and also multiply to equal the number found in Step 3.

Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5.

Step 7: Now that the problem is written with four terms, you can factor by grouping.

Factoring when the leading coefficient is not 1.

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power.

Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together.

Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to $x$.

Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to $x$ and also multiply to equal the number found in Step 3.

Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5.

Step 7: Now that the problem is written with four terms, you can factor by grouping.

Explanation of Solution

Here are some guidelines that you can use to help you factor polynomials. I have listed all of the guidelines first and then I will have some examples of each type. To view examples of each type of factoring problem, click on the links below.

1. Factor out anything that all the terms have in common. This is called the greatest common factor or GCF. This step can either be done at the beginning or the end of the problem. I suggest that you do this step first; it will make the numbers smaller and easier to use.

2. Count the number of terms. Depending on how many terms the problem has, you will use a different factoring technique.

a. Four Terms – If the problem has four terms, you will use a technique called factoring by grouping.

b. Three Terms – If the problem has three terms, you will use a technique that also requires factoring by grouping, but there are a few other steps required to end up with four terms. These steps are shown in detail in the examples. There are two types of examples:

Factoring when the leading coefficient is 1.

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power.

Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together.

Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to $x$.

Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to $x$ and also multiply to equal the number found in Step 3.

Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5.

Step 7: Now that the problem is written with four terms, you can factor by grouping.

Factoring when the leading coefficient is not 1.

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power.

Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together.

Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to $x$.

Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to $x$ and also multiply to equal the number found in Step 3.

Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5.

Step 7: Now that the problem is written with four terms, you can factor by grouping.