When dividing two polynomial expressions, your answer will be a polynomial and is performed like dividing a number, with the main difference being that when determining a term in the quotient you pay attention to the highest power term in the divisor. Take for example the following example:
In this case, the divisor is x + 1 and the dividend is x4 + 3x3 – x2 – x + 2. The term with the highest power in the divisor is the x, so that is the term we will use to determine the quotient. Now, we compare the highest term in the divisor with the highest term in the dividend, x4. Then we determine what we need to multiply the x by to get x4, which is x3. So, we write x3 over the 3x3, as shown below:
Then we multiply the x3 by all the terms in the divisor and write them below the terms with similar powers. Multiplying the x3 by the x gives us x4, so we write it below the x4 and multiplying the x3 by the 1 is x3, and you write it below the 3x3. As seen below:
We then subtract the x4 and x3 from the dividend. x4 minus x4 is 0, and 3x3 minus x3 is 2x3, and then we bring down all the other terms of the dividend, as shown below:
Now we look at the 2x3 and determine what we need to multiply the x by to get 2x3, and that value is 2x2. Writing 2x2 where the quotient goes and multiplying the 2x2 by both the x and 1 gives us 2x3 and 2x2, which we write below the 2x3 and the -x2, as shown below:
Subtracting the 2x3 and 2x2 from the 2x3 – x2 gives us – 3x2 – x – 2.
To get -3x2 we multiply the x by -3x, and multiplying the 1 by -3x is -3x.
Subtracting the -3x2 and the -3x from the 3x2 – x + 2 is 2x + 2.
Multiplying the x + 1 by 2 gives us 2x + 2, and subtracting that from the 2x + 2 provides us with a remainder of 0.
So, x4 + 3x3 – x2 – x + 2 divided by x + 1 is x3 + 2x2 – 3x + 2.