Question

How do you divide `-d^3 + 7d^2 - 11d - 3` by `d+3`?

Answer 1

This can't be done exactly. I didn't work it out below, but if remainders are accepted, then the answer is -d^2 + 10d - 41 with a remainder of 120.

Explanation:

To do this, the expression needs to be able to be factored into something and (d+3).

Honestly, with polynomials of great length like this one, it's best to play around with the equation and see what you can get. Let's try to make that `-d^3` term first.

We can multiply (d+3) by `-d^2` to get `-d^3 - 3d^2`

Now, let's get the next term to be what we want it to be. It should be `7d^2`, and right now it's`-3d^2`, so we need to add `10 d^2`.

`-d^2(d+3) + 10d(d+3) = -d^3 - 3d^2 + 10d^2 + 30d = -d^3 + 7d^2 + 30d`

Last step - make the 3rd term right and see if the last term automatically matches up. If it doesn't then this polynomial can't be divided evenly.

`(-d^2+10d)(d+3) - 41(d+3) = -d^3 + 7d^2 - 11d - 123`

Whoa... that last term isn't right. That means this polynomial isn't divisible by (d+3). You can check this with an online polynomial divider or factorer. You'll find that this gives a remainder and the polynomial doesn't have (d+3) as a factor.