How is factoring a polynomial related to expanding a polynomial

A common method of factoring numbers is to completely factor the number into positive prime factors. A prime number is a number whose only positive factors are 1 and itself. For example, 2, 3, 5, and 7 are all examples of prime numbers. If we completely factor a number into positive prime factors there will only be one way of doing it. That is the reason for factoring things in this way. For our example above with 12 the complete factorization is,.

Factoring polynomials is done in pretty much the same manner. We determine all the terms that were multiplied together to get the given polynomial. We then try to factor each of the terms we found in the first step. This is completely factored since neither of the two factors on the right can be further factored.

Note that the first factor is completely factored however. Here is the complete factorization of this polynomial. The purpose of this section is to familiarize ourselves with many of the techniques for factoring polynomials. The first method for factoring polynomials will be factoring out the greatest common factor. When factoring in general this will also be the first thing that we should try as it will often simplify the problem. To use this method all that we do is look at all the terms and determine if there is a factor that is in common to all the terms.

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.

However, we can still make a guess as to the initial form of the factoring. First, take note of the parentheses present in the equation. Expanding the equation means removing the parentheses.

In order to derive a parentheses-free equation, one simply multiplies the value outside the value, which is 2, to each of the values inside the parentheses. This means that 2 is multiplied to 3c, and 2 is also multiplied to The resulting equation would be 6c Since the equation has no more parentheses, it is said to be completely expanded. If expanding means removing parentheses, then factoring out is the opposite, because it means adding parentheses to an equation.

First, one takes into consideration the common variable between the two values, which is x. Now that the difference between the two terms has been explained, one understands how important it is to know the exact definition of mathematical terms. Knowing how to expand or factor out an equation helps greatly in problem solving. It also enables one to not only solve equations, but also explain objectively the difference between two mathematical terms.

In order to excel at mathematics, one should have a thorough grasp of formulas and mathematical terms. Two commonly used mathematical terms, expanding and factoring, have one thing in common: they deal with either the addition or removal of parentheses in an algebraic equation. Expanding an algebraic equation means getting rid of the parentheses. In order to remove the parentheses, the value outside the parenthesis is multiplied to each of the values inside the parentheses.

On the other hand, factoring out an algebraic equation means adding parentheses to the equation. This is accomplished by taking out the most commonly used value in an equation, then isolating the remaining values in parentheses. Cite APA 7 Franscisco,. Difference Between Expanding and Factoring. Difference Between Similar Terms and Objects. MLA 8 Franscisco,. Name required. Email required. Please note: comment moderation is enabled and may delay your comment.