Divide with long division

Question

(5x³-39x²-49x-45) divided by (x-9)

Answer 1

Polynomial division can be thought of just like normal long division, except the variable "x" must be kept symbolic. In this problem, we would like to find the quotient of two given polynomials, and also the remainder if necessary.

To begin, we write out the given polynomials, (5x³ - 39x² - 49x - 45) and (x - 9). Then, just like long division with numbers, we must ask ourselves, "How many times does (x - 9) go into (5x³ - 39x² - 49x - 45) without going over?" This question is a bit trickier to answer when we have the variable "x" in these expressions.

Let us first look at an example of long division with numbers and then use that as our guide for polynomial division. Take for example, 256 divided by 4. We can see that 4 * 60 = 240. So, we know that 4 goes into 256, 60 times with 256 - 240 = 16 left over. This means that our remainder (so far) is 16. Now we can re-write the expression (256 / 4) = (60 + (256 - 240) / 4) = (60 + 16/4). Then, we can repeat this process again by recognizing that 16/4 is just 4. And so overall, (256 / 4) = (60 + 4) = 64.

Now, let us try this method with the polynomials. The term with the largest exponent in the dividend is 5x³, and the term with the largest exponent in the divisor is x. So, let us multiply (x - 9) by 5x² to get (5x³ - 45x²). Subtracting this from the dividend gives us (5x³ - 39x² - 49x - 45) - (5x³ - 45x²) = (6x² - 49x - 45). So, now we know that our original problem can be re-written as (5x³ - 39x² - 49x - 45) / (x - 9) = 5x² + (6x² - 49x - 45) / (x - 9). Then we can repeat this process until our remainder term is zero, or it cannot be divided into "whole" multiples of (x - 9). We can continue by seeing that (x - 9) * 6x = (6x² - 54x). Subtracting this from the remainder (6x² - 49x - 45) gives us (5x - 45). So now, the answer to the problem is 5x² + 6x + (5x - 45) / (x - 9). Next, we will multiply (x - 9) by 5 to get (5x - 45). But this is the same as the last remainder, so now our remainder is zero! That means we are finished, and the answer is (5x³ - 39x² - 49x - 45) / (x - 9) = (5x² + 6x + 5).