In algebra problems, you will often have to combine polynomials with addition or subtraction. These problems can look confusing, but they are fairly simple. You just have to do them carefully, paying particular attention to negative signs.

There is an associative property of addition, so you can remove the parentheses and just combine the like terms from each polynomial.  Pay chose attention to:

• Variables
• Exponents on variables (remember: squared terms can only be combined with other squared terms of the same variable)
• Positive and negative signs (combine like terms with either addition or subtraction, depending on the signs on the terms)

Example:

$(2x^2+6x-9)+(4x^2+5x+10)$

The best way to do these kinds of problems is to work from the highest power terms to the lowest, circling like terms, combining them, then crossing them out before moving onto the next like terms.

\require{cancel}\eqalign{(2x^2+6x-9)+(4x^2+5x+10)&=\\2x^2+6x-9+4x^2+5x+10&=\qquad&&\text{Remove parentheses}\\\boxed{2x^2}+6x-9\boxed{+4x^2}+5x+10&=&&\text{Circle like terms}\\\boxed{\cancel{2x^2}}+6x-9\boxed{\cancel{+4x^2}}+5x+10&=6x^2+6x-9+5x+10&&\text{Combine like terms}\\6x^2\boxed{+6x}-9\boxed{+5x}+10&= &&\text{Circle next like terms}\\6x^2\boxed{\cancel{+6x}}-9\boxed{\cancel{+5x}}+10&=6x^2+11x-9+10\\6x^2+11x\boxed{-9+10}&=&&\text{Combine last like terms}\\6x^2+11x\boxed{\cancel{-9+10}}&=\mathbf{6x^2+11x+1}}

Other times, you have to subtract polynomials.  The process is exactly the same, but before you start to combine like terms, distribute the negative sign onto each term inside any subtracted polynomial.

Example:

$(2x^2+4x-9)-(4x^2+5x-2)$

Distribute the negative sign (making positive terms negative and negative terms positive) then combine like terms just like you did in the problem above.

\eqalign{(2x^2+4x-9)-(4x^2+5x-2)&=\\2x^2+4x-9-4x^2-5x+10&=\qquad&&\text{Distribute - on terms in subtracted polynomial}\\\boxed{2x^2}+4x-9\boxed{-4x^2}-5x+2&=&&\text{Circle like terms}\\\boxed{\cancel{2x^2}}+4x-9\boxed{\cancel{-4x^2}}-5x+2&=-2x^2+4x-9-5x+2&&\text{Combine like terms}\\-2x^2\boxed{+4x}-9\boxed{-5x}+2&= &&\text{Circle next like terms}\\-2x^2\boxed{\cancel{+4x}}-9\boxed{\cancel{-5x}}+2&=-2x^2-x-9+2\\-2x^2-x\boxed{-9+2}&=&&\text{Combine last like terms}\\-2x^2-x\boxed{\cancel{-9+2}}&=\mathbf{6x^2-x-7}}

As long as you carefully combine all of the like terms, and pay attention to negative signs, you can easily add polynomials of any size!

Practice Problems:

Simplify the following expressions:

1. $(7b^2-7b-10)+(9b^2-3b-5)$

2. $(8a^2+10a-15)+(9a^2-9b+10)$

3. $(6x^2-7x+10)-(4x^2+8x+8)$

4. $10b-b^2+3b-10+b^2$

5. $(c^3+2c^2+c+1)+(4c^3+5c-3)$

6. $(x^4+x+15)-(x^3-x^2-7x+18)$

7. $(-17z-89)+(z^2+6z-10)$

8. $(3c^2+7n+8)-(6t^2+6c-3n+4)$

9. $(6x^3+62)-(x^3-x^2-x-92)$

10. $(7x^2-8x-9y+19)-(6x^2-9y^2-8x+7y)$

11. $(6g^2+5g-9p+9)+(-2g^2-6g-2)$

12. $(6y+7x+10)-(6y^2+4x-20)$