For this post, I would like to do one that talks about various methods of factoring which you can use to find factors. There are several different factoring techniques that you can use, depending on the situation, so I think that I would like to include a variety of methods or factoring tricks.
First of all, a high level, general definition is needed. What does factoring mean? Factoring means to simplify a mathematic expression by writing the expression as a product of two or more values or expressions. For example, if you factor the expression 12x + 4, you get (4)(3x+1). This is an example of “factoring out” the 4. You will see this phrase “factoring out” very commonly when dealing with these types of problems.
Several factoring techniques are available to you to help you find factors, depending on the question. These are some techniques you can use. You can memorize these factoring tricks and shortcuts and you will save a lot of time with your math work! If you do lots of practice with factoring games or math worksheets, you will become good at these and be able to find factors very quickly.
In an expression composed of multiple terms, try to identify if there is one number/variable that is a common factor to each term. Then, after factoring out the common factor, you can rewrite the expression to show multiplying that common factor by the remaining terms.
12x + 4 = (4)(3x + 1)
5x3 + 10x2 + 25x = (5x)(x2 + 2x + 5)
Difference of squares
Wherever you see a difference of two terms that are perfect squares (either something like x2 or 25), you can apply this technique for factoring a difference of two squares. This actually is the same as one of the rules for special polynomial products. In fact, factoring special products follow the same rules you would use to find factors anywhere else. Factoring squares is actually quite simple:
x2 – a2 = (x – a)(x + a)
You reduce the terms to their square root value, and remember to put one ‘+’ and one ‘-’. Easy as that.
Difference of cubes / Sum of cubes
Factoring perfect cubes (or factoring cubics) is a little trickier, but they follow a strict form that you can memorize and use easily to find factors. Depending on if you are subtracting perfect cubes or adding perfect cubes, you will use the appropriate formula:
x3 – a3 = (x – a)(x2 + ax + a2)
x3 + a3 = (x + a)(x2 – ax + a2)
They have the same basic form, you just have to pay attention to the signs. Don’t mix them up!
This is an extension of the common factor method described above. The only difference is that the common factor doesn’t have to be common with EVERY term. Group things together, and factor within the groups. Take this example and you should see what I mean:
x3 – x2 + x –1
= (x3 – x2) + (x –1)
= x2(x – 1) + (x – 1)
= (x – 1) (x2 + 1)
Study that example and it should be fairly self-explanatory how I used grouping to find factors here.
Trial and Error
Sometimes, there aren’t any obvious factoring tricks or factoring techniques that you can apply to help you solve your question. Unfortunately, in these situations, you must resort to trial and error. Sometimes you can figure out the numbers that are involved, but you need to test out the signs to get it right. I can’t really say anything about this technique except to have patience and keep trying. 🙁
I will revisit this post shortly to put up a remark about factoring quadratic equations. I will also go over AC method factoring. The AC method of factoring is a factoring method or factoring trick you can use to help you factor expressions in the form of ax2 + bx + c (trinomial).