8.3.2 Synthetic Division
To divide a polynomial by another polynomial, we use the same process of long division that we would use with regular numbers.
In the previous section, we asked about how we can divide something that looks like: \frac{2x^2-3x+5}{x+2}
Eventually we'd like to be able to study equations that involve rational expressions but at this point, we're just getting to know these new mathematical creatures.
The tools that we've used so far to simplify them are:
- Factoring out and then canceling.
- Breaking our expression into smaller pieces and then canceling some of them.
Unfortunately, neither technique works with
\frac{2x^3-3x+5}{x+2}
so we're going to have develop a new technique.
What other technique do we have to divide numbers?
Let's take a look at at example. Suppose we want to divide 16 into 225.
\longdiv{225}{16}
Explore!
If you divide 7 into 942, then it divides times with a remainder of
You can see a full explanation below:
\longdiv{942}{7}
Since polynomials are just patterns of numbers, we can use the same exact process in the more abstract setting:
\polylongdiv{2x^3+x^2-x}{x+2}
Explore!
Dividing by x-3 gives us a remainder of
You can see the full explanation below:
\polylongdiv{x^4+x^2+1}{x-3}
Division of polynomials seems like a very abstract concept. Are there any phenomena in the real world that we can model by rational expressions? We'll study some in the next section.
