Why is factoring polynomials important

How to Factor Polynomials Step-by-Step. How to Use Significant Figures in Multiplication and How to Convert Nanograms to Milligrams. How to Factorise a Quadratic Expression. What Is the Square Root Method? How to Figure Survey Percentages. How to Factorise in Math. The Foil Method With Fractions. How to Factor Expressions in Algebra. If there is, we will factor it out of the polynomial. Also note that in this case we are really only using the distributive law in reverse.

Remember that the distributive law states that. In factoring out the greatest common factor we do this in reverse. First, we will notice that we can factor a 2 out of every term.

Here then is the factoring for this problem. Note that we can always check our factoring by multiplying the terms back out to make sure we get the original polynomial.

Doing this gives,. Remember that we can always check by multiplying the two back out to make sure we get the original. Be careful with this. This one looks a little odd in comparison to the others.

However, it works the same way. Doing the factoring for this problem gives,. This method is best illustrated with an example or two. Note again that this will not always work and sometimes the only way to know if it will work or not is to try it and see what you get. This gives,. So we know that the largest exponent in a quadratic polynomial will be a 2. In these problems we will be attempting to factor quadratic polynomials into two first degree hence forth linear polynomials.

Until you become good at these, we usually end up doing these by trial and error although there are a couple of processes that can make them somewhat easier. To finish this we just need to determine the two numbers that need to go in the blank spots. We can narrow down the possibilities considerably. Upon multiplying the two factors out these two numbers will need to multiply out to get In other words, these two numbers must be factors of Here are all the possible ways to factor using only integers.

Now, we can just plug these in one after another and multiply out until we get the correct pair. However, there is another trick that we can use here to help us out. Now, we need two numbers that multiply to get 24 and add to get It looks like -6 and -4 will do the trick and so the factored form of this polynomial is,. This time we need two numbers that multiply to get 9 and add to get 6.

In this case 3 and 3 will be the correct pair of numbers. Note as well that we further simplified the factoring to acknowledge that it is a perfect square. You should always do this when it happens. Okay, this time we need two numbers that multiply to get 1 and add to get 5. When are we happiest? When there's no difference:. If we have a system and the desired state, we can make a new equation to track the difference -- and try make it zero.

This is deeper than just "subtract 6 from both sides" -- we're trying to describe the error! Factoring the rescue. Imagine taking a pile of sticks our messy, disorganized system and standing them up so they support each other, like a teepee:.

If Component A or Component B becomes 0, the structure collapses, and we get 0 as a result. That is why factoring rocks: we re-arrange our error-system into a fragile teepee, so we can break it. We'll find what obliterates our errors and puts our system in the ideal state. I've wondered about the real purpose of factoring for a long, long time.

In algebra class, equations are conveniently set to zero, and we're not sure why. Here's what happens in the real world:. The idea of "matching a system to its desired state" is just one interpretation of why factoring is useful. If you have more, I'd like to hear them! Multiplication is often seen as AND. If either condition is false, the system breaks. The Fundamental Theorem of Algebra proves you have as many "components" as the highest polynomial.