Question

How do you add `(2x^2 - 4xy + y^2) + ( -6x^2 - 9xy - y^2) + (x^2 + xy - 6y^2)`?

Answer 1

`-3x^2 - 12xy - 6y^2`

Explanation:

When adding terms in polynomials, you are allowed to combine "like terms".

Two terms are "like" if they have the same variable(s) with possibly different constant coefficients. Recall that in a term like `17x^2`, the number `17` is the coefficient and the `x^2` is the variable.

For some examples, `6x^2` and `8x^2` are "like terms", because their variable parts are both `x^2`. Also, `xyzh^2` and `-13xyzh^2` are also like, since their variables are both `xyzh^2`. The terms `3y` and `3xy` are not like terms, because one has a variable of `x` and the other of `xy`.

If you have two "like terms", you add them (or subtract them) by adding (or subtracting) their coefficients. So `2x^2 + 5x^2 = 7x^2`. Similarly, `4xy - 7xy = -3xy`.

Now to the problem at hand. We have:

`2x^2 - 4xy + y^2 - 6x^2 - 9xy - y^2 + x^2 + xy - 6y^2`

To make things more clear, we'll rearrange the terms (keeping their positive or negative signs) so that like terms are next to one another.

`(2x^2 - 6x^2 + x^2) + (- 4xy - 9xy + xy) + (y^2 - y^2 - 6y^2)`

Now we just add the coefficients.

`(2 - 6 + 1)x^2 + (-4 - 9 + 1)xy + (1 - 1 - 6)y^2`
`= -3x^2 - 12xy - 6y^2`

Now there are no two like terms, meaning we are done adding. This is our final answer.