The most important thing that I have learned this week in Math 10 and by the end of the polynomials and factoring unit, is how to completely factor a polynomial. Completely factoring a polynomial is to continue to factor it until it is at it’s most expanded form possible The expanded form of a polynomial expression is made up of its factors, which are the numbers that can be multiplied by each other to result in the polynomial we started with, and the complete factorization is the prime factors, meaning that they can’t be reduced any further to more factors. It is very useful to know how to factor a polynomial completely because it can help to break down expressions, and knowing the different ways that polynomials may need to be factored allows you to be able to factor a greater variety of algebraic expressions. The reason that I chose this as the most important thing that I have learned is because there are three ways to factor polynomials that you can go through like a checklist, and once you have factored one way, you see if the next one can be applied to know if you are done factoring or not; this can be remembered as factoring 1-2-3. This was an extremely useful strategy for factoring, and it is one of the main reasons that I chose this topic because it is useful to remember and can be applied in the future.

How to completely factor a polynomial using factoring 1-2-3:

Step 1:

The first thing to look for when factoring a polynomial is to see if there is a greatest common factor between all of the terms in the expression. The greatest common factor is the highest number and/or variable that can be multiplied into each term in the expression. Once you have identified the greatest common factor, you can remove it by moving it to the outside of the brackets and dividing the inside terms by it. This way you have separated the terms into factors.

Ex.

 

Step 2:

Even after you have removed the greatest common factor, it is possible that the expression can still be factored further; the next way to factor a polynomial, or thing to check for, is if it is a difference of squares. This is for a polynomial with two terms, or a binomial. If a binomial is a difference of squares, both values will be perfect squares, and the sign in between will be a minus. To factor these, you take the square roots of each term in the binomial, and put them in a binomial together, to be multiplied by a binomial with the same terms inside, except with the opposite operation, so one will be an addition and one will be a subtraction, to result in the binomial that we began with, which was a subtraction.

Ex.

 

Step 3:

The final part of factoring 1-2-3 is for trinomials and is specific to trinomials of the form . In trinomials like these, the middle term is the sum of the two number factors and the last term is the product of the two number factors. The factorization is a binomial, and each term consists of an x and one of the two number factors. You can use this knowledge to be able to find the factors for a trinomial of this form, and it is one of the things that you should check for to know how you can proceed with factoring the expression.

Ex.

 

Here is an example of completely factoring a polynomial where you have to use more than one step:

Step 1: Using factoring 1-2-3, the first thing we would check for is if there is a greatest common factor between the two terms. In this example, the re is a greatest common factor which is 2, so we would start by removing it.

Step 2: In this example, the expression can still be further factored. We now have 2 multiplying a binomial made up of two perfect squares and separated by a minus sign, so it is a difference of squares and can be factored as such.

This example is now completely factored and used two of the elements of factoring 1-2-3.