What is factor e^{-2t}u(t)+e^{3t}u(-t) ?

The solution to factor e^{-2t}u(t)+e^{3t}u(-t) is -e^{-2t}tu(e^t-1)(e^{4t}+e^{3t}+e^{2t}+e^t+1)

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factor e^{-2t}u(t)+e^{3t}u(-t)

factor

e^{- 2 t} u (t) + e^{3 t} u (- t)

- e^{- 2 t} t u (e^{t} - 1) (e^{4 t} + e^{3 t} + e^{2 t} + e^{t} + 1)

Solution steps

e^{- 2 t} u (t) + e^{3 t} u (- t)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

e^{3 t} = e^{- 2 t} e^{5 t}

= e^{- 2 t} u t + e^{- 2 t} e^{5 t} u (- t)

Factor out common term

u t e^{- 2 t}

= u t e^{- 2 t} (1 - e^{5 t})

Factor

- e^{5 t} + 1 : - (e^{t} - 1) (e^{4 t} + e^{3 t} + e^{2 t} + e^{t} + 1)

= e^{- 2 t} t u (- (e^{t} - 1) (e^{t} + e^{4 t} + e^{3 t} + e^{2 t} + 1))

Refine

= - e^{- 2 t} t u (e^{t} - 1) (e^{t} + e^{4 t} + e^{3 t} + e^{2 t} + 1)

x^{2} - 12 + 35

(a + b)^{3} - (a - b)^{3}

125 a^{3} + 8 a^{3}

y^{3} - 64 a^{3} b^{3}

x^{2} - 11 x + (- 28)

What is factor e^{-2t}u(t)+e^{3t}u(-t) ?
The solution to factor e^{-2t}u(t)+e^{3t}u(-t) is -e^{-2t}tu(e^t-1)(e^{4t}+e^{3t}+e^{2t}+e^t+1)