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What do you do with the exponent when adding, say, p^2 and -p^3? What are the steps to finding the answer? DO NOT just TELL me the answer. This is just an example. I need to know how it works so I can solve the real equation.

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The exponents would have to be the same to add, with you example the final answer is p^2-p^3.
EDIT: multiplication is a different story though, with multiplying you would add the exponents together.

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Thanks. So they aren't like terms? what about if you expand them? then what do you do? And, uh, what do you do if both coefficients are negative? I know that if you had (-p)^2 it could be -p^2(-p^2), so I assume that -p^2 would be -p(p), am i right?
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-p^2 means -p x -p
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No, -p^2 = -p x p
(-p)^2 = -p x -p = p
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you can take common factor out.

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I am adding, not multiplying. And by "common factor," do you mean coefficient?
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even you are adding, the common factor can be taken out. say for example: 3^2-3^3 = 9 - 27 = -18
or
3^2(1-3) = 9(1-3) = 9(-2) = -18
you can notice here we need to take smaller power term out.
this method is required when you go for cancelling terms during divisions of polynomials.
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The exponents have to be the same to add. If they are then add the coefficients.
Examples:
p^2 + -p^3 = p^2 - p^3
2x^5 + 7x^5 = 9x^5
Multiply monomials
Add the exponents. Multiply the coefficients.
Examples:
3x^4 x 2x^3 = 6x^7
p^2 x -p^3 = -p^5