What is expand (x+Δx)^4 ?

The solution to expand (x+Δx)^4 is x^4+4Δx^4+6Δ^2x^4+4Δ^3x^4+Δ^4x^4

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expand (x+Δx)^4

expand

(x + Δ x)^{4}

x^{4} + 4Δ x^{4} + 6 Δ^{2} x^{4} + 4 Δ^{3} x^{4} + Δ^{4} x^{4}

Solution steps

(x + Δ x)^{4}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = Δ x

= i = 0 \sum 4 (i 4) x^{(4 - i)} (Δ x)^{i}

Expand summation

= \frac{4 !}{0 ! ( 4 - 0 ) !} x^{4} (Δ x)^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} x^{3} (Δ x)^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} x^{2} (Δ x)^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} x^{1} (Δ x)^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} x^{0} (Δ x)^{4}

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} x^{4} (Δ x)^{0} : x^{4}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} x^{3} (Δ x)^{1} : 4Δ x^{4}

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} x^{2} (Δ x)^{2} : 6 Δ^{2} x^{4}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} x^{1} (Δ x)^{3} : 4 Δ^{3} x^{4}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} x^{0} (Δ x)^{4} : Δ^{4} x^{4}

= x^{4} + 4Δ x^{4} + 6 Δ^{2} x^{4} + 4 Δ^{3} x^{4} + Δ^{4} x^{4}

\frac{n ^{2} ( n + 1 ) ^{2}}{4}

(- 1 + 3 i)^{4}

\frac{1}{( s + 1 ) ^{2}}

\frac{1}{x ( x + 1 )}

(cos (θ) + i sin (θ))^{3}

What is expand (x+Δx)^4 ?
The solution to expand (x+Δx)^4 is x^4+4Δx^4+6Δ^2x^4+4Δ^3x^4+Δ^4x^4