Question

How do you simplify `(5r ^ { 3} - 3r ^ { 2} + 6) + ( 3r ^ { 3} + 4r ^ { 4} + 8r ^ { 2} )`?

Answer 1

Remove unnecessary brackets, "convert" subtraction to negative addition (to prevent miscalculations), rearrange, and add together like terms, to get `4r^4 + 8r^3 + 5r^2 + 6`.

Explanation:

So we have:

`(5r^3 - 3r^2 + 6) + (3r^3 + 4r^4 + 8r^2)`

One might think of first performing the operations inside the brackets, but note that it is addition, so the brackets don't really matter here:

`5r^3 - 3r^2 + 6 + 3r^3 + 4r^4 + 8r^2`

Now, a conventional "arrangement" of polynomials is to sort their terms, left to right, from the highest degree (exponent) to the lowest degree (which could be a constant).

Before rearranging terms, however, I'd usually make sure all subtractions are "converted" into negative additions (and divisions to reciprocal multiplications), not only to prevent miscalculations, but to also "use the full power" of addition (and multiplication) commutativity (the ability to swap terms) here. Let's do that for `-3r^2`:

`5r^3 + (-3)r^2 + 6 + 3r^3 + 4r^4 + 8r^2`

Back to rearranging terms, we have `r^4` being the highest power here, so move that (along with its coefficient!) over to the left first:

`4r^4 + 5r^3 + (-3)r^2 + 6 + 3r^3 + 8r^2`

The next is `r^3`, and we have `5r^3` and `3r^3`:

`4r^4 + 5r^3 + 3r^3 + (-3)r^2 + 6 + 8r^2`

Since these are like terms, we can add them! `5` of something plus `3` of that thing is `8` of that same thing.

`4r^4 + 8r^3 + (-3)r^2 + 6 + 8r^2`

Then, `r^2`:

`4r^4 + 8r^3 + (-3)r^2 + 8r^2 + 6`

We can add `-3` and `8`, to become `5`:

`4r^4 + 8r^3 + 5r^2 + 6`

What's next would be `r^1`, which is not there, and `r^0`, the constant, but that's already in place, so that's our answer!