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Factorization

Exponential Functions

Two extra tricks need to be kept in mind when dealing with exponential functions.

e^nx = (e^x)ⁿ

e^{a + b} = (e^a)(e^b)

Example 5
4xe^2x 8e^x	Strategy
4xe^2x and 8e^x	Consider each product separately, but watch out for sums which may be in common
4 and 8 have a HCF of 4	Find the highest common factor of the numeric parts of the product expressions.
x is in only one term	Find the common factor for factors involving a variable.
Since e^2x = (e^x)² e^2x and e^x have e^x in common	Find the common factor for factors which involve exponential factors.
Hence 4e^x is the largest common factor.
4xe^2x 8e^x = 4e^x(xe^x) + 4e^x(2) = 4e^x(xe^x + 2)	Put the parts together.

Here are some exercises to finish off:

Factorise these then check your answers.

e^3x - e^2x

4e^x + e^4x

e^{y + 2} + e^2y