FUNdamental Factoring

What is Factoring?

Factoring is the process of finding factors. It can sometimes be difficult to factor because you have to figure out what has gotten multiplied to produce the expression you are given. In more basic terms, factoring is the opposite of expanding. Therefore it is alike 'splitting' an expression into a multiplication of similar expressions.

Factoring Tips

-First factor out any common terms.

-See if the expression fits any of the following identities and factor using that method. (Methods includes: Common Factoring,  Sum and Product, Grouping, Decomposition,  Difference of Squares)

-Continue following that method until you can't factor anymore. In order for you to be unable to factor anymore the expression must be in simplest terms.

Now you may ask...

Factoring is a basic skill that is applied through many other math scenarios and certain careers in the the workplace. One of these careers is an engineer. If you think about how control circuitry is modeled as different equations, such as a thermostat, it can give engineers an idea about how efficient the system is. The next step would be finding roots of these polynomials within the system in order to describe the structure better. This is one of many other examples when it comes to engineering.

Another more relatable example regarding factoring could be using cruise control or autopilot. It is important to be able to monitor and test these functions in order to make sure the system is running properly. This is possible by using factoring because such as the engineering example, people are able to go through the roots of the system in become more knowledgeable about the systems.

In the previous paragraphs 'the next step' was stated to be finding roots. This is in relation to quadratics which we need to solve problems not only in math, but also life. Think about an apple orchard for example; the fewer apples you get per tree the more crowded to orchard is bound to be depending on the amount of apples you want to produce. This can be found through a quadratic equation which to solve we first need the initial skill of factoring. As pointless as factoring may seem at times, it can lead to finding the solution for many daily problems!

Methods of Factoring

Common Factoring

When common factoring you must look for the highest common factor (including variables!) This will help you to find the fully simplified solution. Check out the examples below.

EXAMPLE #1:

In this example the highest common factor was found to be 3. This is clear when you look for similarities between the numbers, but also through the second line. When looking at this line you see the common factor and (2x-1). Now you may wonder: Where did (2x-1) come from? Well...

When the common factor of 3  is multiplied into (2x-1), the answer produced is (6x-3), which is the initial question. -->  3*2x= 6x   &    3*-1= -3

Therefore you must do the opposite of what you've previously done to the expression in order to find the lowest expressed factor. In this case the opposite would be division because (6x-3) divided by 3 equals (2x-1).  

EXAMPLE #2:

By using the explanation in EXAMPLE #1 see if you can identify the common factor... If you guessed 4x^2 you are correct! Now if you divide that factor out of the original question, your answer should be identical to the one above.  --> 4x^5 / 4x^2 = x^3    &    28x^2 / 4x^2 = 7

Sum and Product

When completing sum and product questions you must look for the following properties within the question: The expression is a trinominal, the 'z' or other variable is equal to one, and that two specific numbers add up to 'x' and multiply to 'y'. (Format: z+x+y)

EXAMPLE #1:

To solve this example there are a couple things you need to do. It makes it easier if you identify the sum and product first. The sum will always be the number representing 'x' and the product will be the number representing 'y' as shown in the format above. (Please note that your question may not always be in this order.) So, in this case the sum is 14 and the product is 48.

Now that you are aware of all the numbers you need, you must find two numbers that multiply to the product and add to the sum. If you are unsure of what the pair of numbers might be, sometimes it is easier to make a list of  multiples for the product. For example: 1,48; 2,24; 3,16; 4,12; 6,8.

Now, because 6+8=14 and 6*8=48 we have found the numbers we are looking for. Therefore our answer will be (x+6)(x+8) because if you follow FOIL your answer will then be the same as the initial question. In this example the First two numbers within the brackets will give you x^2, the Outer and Inner numbers will give you 14x, and the Last numbers will give you 48. If you follow this method when putting together your answers it will make it easier to find the solution to the question.

EXAMPLE #2:

At first glance this may not look like a sum and product question because the first variable isn't equal to one. Although if you are able to equally divide out that variable, I would very much suggest it. In this case 3 was divisible by each of the numbers as shown within the second line. -> 21x/3=7x    &   30/3=10

Now, can you identify the sum and product by looking at the second line? If you said -7 was the sum and 10 was the product you are correct! Although this question is slightly different due to the negative,  this makes very little difference as you are still trying to achieve the same goal of finding the right pair of numbers. Which in this case is -2 and -5. You then use the FOIL method as described earlier to produce your answer. Remember to keep any number you may have divided out to begin with or else your answer will be unequal to the question.

Grouping

When grouping you are usually looking for similarities between 4 terms presented within an expression. Although there can be several different ways to group an expression all answers should result to be the same.

EXAMPLE #1:

In this example you may have noticed that there are two different ways of solving the same question. (Note: The second line in both questions has been rearranged in order to see the grouping process better.) When grouping you only compare two terms at a time, but each of the brackets should result to be equal. In the first example 'x' and 'y' were divided out of their original spots while in the second one 'a' and 'b' were. Notice that no matter what variables you chose to divide out, the brackets shown in both third lines are equal. These can then be simplified into the answers shown above because in both cases you are multiplying the same brackets by either (x-y) or (a+b).

EXAMPLE #2:

This example is a little more tricky when it comes to grouping due to the extra number, although it is very easy once you see the FOIL pattern. Within the second line 4 is divided out of x and y which equals 4(x+y). This makes the equation easier to factor because it is more simple to see where the (x+y)^2 comes from as well as 4x+4y. This is seen in the third line because once FOILed it will give you an answer equivalent to the question.

Decomposition

When you are finishing decomposition questions you are looking for the following properties: The expression is a trinominal, the 'z' or other variable is NOT equal to one, and that you are able to FOIL the final solution to recreate the initial question. (Format: z+x+y) If you are unsure of what FOIL means it stands for First, Outer, Inner, Last... this is the process in which brackets are expanded.

EXAMPLE #1:

When completing a decomposition question you must also identify the sum and product. One difference is that you must multiply 'y' and 'z' as shown in the format above. This will give you your product, which is 6 in this case because 2*3=6. Your sum is still going to be 'x' which is 7 in this example. At this point you must find the proper pair of numbers that equal the product and sum. Instead of creating your brackets in the next line like sum and product questions you write down those numbers in between your 'y' and 'z' numbers followed by a variable. This creates the second line.

After this step you can now group the rest of the question in order to find the solution. If you are unsure of how grouping works, please look at the section above.

EXAMPLE #2:

Within this example you will follow the same steps as in EXAMPLE #1. So, when you multiply 'y' and 'z'; which is -> 5*-20=-100 <- it will give you the product. The sum must then be -48. If you then go though the possible multiples of -100, it is clear that the pair of numbers must be -50 and 2; this is written on the second line. After this line you then must follow the grouping steps by finding similarities between the first two numbers and the last two. If you need additional grouping help check out the section above.

Difference of Squares

When doing difference of squares you are looking for binomials, a negative sign, and two perfect squares. Sometimes it is necessary to look deeper into the question as these properties aren't always clearly visible (such as in example#2).

EXAMPLE #1:

To find the answer for difference of squares questions you must also follow the FOIL method. First find the pair of numbers that create each square within the question. This can be found if you square root each of the original numbers.  Therefore the squares within the question are ->  y*y=y^2   &   -6*6=-36 <- and they create up the final answer of (y-6)(y+6).

EXAMPLE #2:

Like stated before some questions aren't clearly visible as difference of squares questions, but like in any other questions such as this you are looking for the numbers that create the squares. In the second line of this example you see that this was done because the square root of (x-1)^2 is (x-1) and the square root of 9 is 3 (Note: Due to the fact that 9 is a negative in the question one of the 3's must also be negative.) The squares are just combined within each of the brackets in order to produce (x-1+3)(x-1-3). The final answer is then simplified creating our final answer of (x+2)(x-4).

REMEMBER:
Experience and Patience Make All The Difference When You're Factoring!

If you follow these simple steps your factoring skills will develop to be very strong.

Good Luck In Your Studies!

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