Division Sometimes fractions are a pleasure to divide such as 4/2 or 100/10. All we have to do is divide the denominator into the numerator to get 2 and 10 respectively for the fractions we just named. Sometimes dividing fractions is a bit more involved such as when dividing polynomials. The prefix 'poly' means 'many' and 'nominal' means numbers or 'terms'. Let's focus on how to divide polynomials with two variables. Two Variable Polynomial Division Example 1 Let's look at the fraction We want to evaluate it. In other words, what polynomial times the denominator ( x 2 + y) will give the polynomial in the numerator?

1. Manual Synthetic Division Of Polynomials By Trinomials
2. Manual Synthetic Division Of Polynomials Examples
3. Synthetic Division Steps

Manual Synthetic Division Of Polynomials By Trinomials

The process is very similar to numerical long division. Let's see how to do this. The first step is to rewrite the problem giving us Some term that we will put in the red box times x 2, which is circled in red, has to give us x 3, which is outlined in green. The term we are looking for is x, which we will put in the red box. We now multiply x by x 2 and by the y term: x times x 2 gives x 3 and x times y gives xy.

We subtract these two terms from the original polynomial in the numerator. Now we go to the next term, x 4 y. What term would go in the next red box that when multiplied by ( x 2) will give us x 4 y? Well, x 2 y times x 2 gives us x 4 y. We also have to multiply x 2 y by y giving us x 2 y 2. We subtract these two terms from the original numerator. Let's see what that looks like.

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Manual Synthetic Division Of Polynomials Examples

The next term in the polynomial we have to work with is x 3 y 2. What times x 2 gives us x 3 y 2? The answer to that question is xy 2, which we multiply by both x 2 and by y. Subtracting these terms from the original numerator leaves us with no other terms that need to be worked with, which means we have our answer! It is To check if we did this work correctly we can multiply the original denominator ( x 2 + y) by our answer. If the result is our original numerator, we did everything correctly! First we set up the terms to be multiplied.

We take the x 2 and multiply it by every term in our answer. This results in Now we multiply the second term from our original denominator ( y) by our answer resulting in Adding these together we get This is the numerator we started with, which means we did everything correctly! Let's do one more example. Example 2 Divide into Setting this up we get.

Now we need to find a term that when multiplied by x 3 gives us x 7 z. The term that will do this is x 4 z. Multiplying this by ( x 3 - z) we get x 7 z and ( -x 4 z 2). Subtracting these terms from what's inside the original long division sign we get The next term we need to work with is ( -x 3 z 5). What times ( x 3 - z) gives us this? The answer is -z 5.

Multiplying this by ( x 3 - z) gives us ( -x 3 z 5) and z 6. Putting in these terms into our long division gives us Our answer is Checking our answer gives us Since we ended up with what we started with we know we did all of the work correctly!

Lesson Summary A polynomial is a mathematical expression with multiple terms. Dividing polynomials with two variables is very similar to regular long division. We go through each term of the polynomial determining what goes into it and subtracting that term from the original polynomial.

Synthetic Division Steps

When there are no terms left in the original polynomial the division is complete. You can check your answer by multiplying it by the polynomial that is dividing into the polynomial answer you got. If you end up with what you started with you have done everything correctly.