What is expand (1+te^{-t})^3 ?

The solution to expand (1+te^{-t})^3 is 1+3e^{-t}t+3e^{-2t}t^2+e^{-3t}t^3

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expand (1+te^{-t})^3

expand

(1 + t e^{- t})^{3}

1 + 3 e^{- t} t + 3 e^{- 2 t} t^{2} + e^{- 3 t} t^{3}

Solution steps

(1 + t e^{- t})^{3}

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

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

a = 1, b = t e^{- t}

= 1^{3} + 3 \cdot 1^{2} t e^{- t} + 3 \cdot 1 \cdot (t e^{- t})^{2} + (t e^{- t})^{3}

Simplify

1^{3} + 3 \cdot 1^{2} t e^{- t} + 3 \cdot 1 \cdot (t e^{- t})^{2} + (t e^{- t})^{3} : 1 + 3 e^{- t} t + 3 e^{- 2 t} t^{2} + e^{- 3 t} t^{3}

= 1 + 3 e^{- t} t + 3 e^{- 2 t} t^{2} + e^{- 3 t} t^{3}

(x - 4) (x - 6) (x - 7) (x + 7)

(x - 2 y - 1) d x + (3 x - 6 y + 2) d y

(2 x + 1) y^{''} (x) + x y

\frac{d}{d y} ((f (y, x)) (x^{2} y + x + x y^{2} + y))

(x - 1) (x + 5) (x - 5)

What is expand (1+te^{-t})^3 ?
The solution to expand (1+te^{-t})^3 is 1+3e^{-t}t+3e^{-2t}t^2+e^{-3t}t^3