Question

Question #43bd9

Answer 1

`(e^x + e^-x)^2`

Explanation:

I'm assuming you mean `e^(2x) + 2 + e^(-2x)`.

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There's something very special about this polynomial. To see it, let's use one of the rules of exponents to write the first and last terms in a different way:

`a^(bc) = (a^b)^c`

Therefore, we can change our polynomial to look like this:

`e^(2x) + 2 + e^(-2x)`

`(e^x)^2 + 2 + (e^-x)^2`

Also, remember that `e^-x` is the same as `1/e^x`.

`(e^x)^2 + 2 + (1/e^x)^2`

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Do you see it? This polynomial is actually a perfect square! Remember this formula?

`(a+b)^2 = a^2 + 2ab + b^2`

Well, if we use the fact that `(e^x)(1/e^x) = 1`, we can change our polynomial to look like this:

`(e^x)^2 + 2(e^x)(1/e^x) + (1/e^x)^2`

This very clearly fits with our perfect square formula, so we can factor it like this:

`(e^x + 1/e^x)^2`

Or, to write it like the original problem did,

`(e^x + e^-x)^2`

Final Answer