What is expand (x+Δx)^6 ?

The solution to expand (x+Δx)^6 is x^6+6Δx^6+15Δ^2x^6+20Δ^3x^6+15Δ^4x^6+6Δ^5x^6+Δ^6x^6

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expand (x+Δx)^6

expand

(x + Δ x)^{6}

x^{6} + 6Δ x^{6} + 15 Δ^{2} x^{6} + 20 Δ^{3} x^{6} + 15 Δ^{4} x^{6} + 6 Δ^{5} x^{6} + Δ^{6} x^{6}

Solution steps

(x + Δ x)^{6}

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 6 (i 6) x^{(6 - i)} (Δ x)^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} (Δ x)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} (Δ x)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} (Δ x)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} (Δ x)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} (Δ x)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} (Δ x)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} (Δ x)^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} (Δ x)^{0} : x^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} (Δ x)^{1} : 6Δ x^{6}

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} (Δ x)^{2} : 15 Δ^{2} x^{6}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} (Δ x)^{3} : 20 Δ^{3} x^{6}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} (Δ x)^{4} : 15 Δ^{4} x^{6}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} (Δ x)^{5} : 6 Δ^{5} x^{6}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} (Δ x)^{6} : Δ^{6} x^{6}

= x^{6} + 6Δ x^{6} + 15 Δ^{2} x^{6} + 20 Δ^{3} x^{6} + 15 Δ^{4} x^{6} + 6 Δ^{5} x^{6} + Δ^{6} x^{6}

(6 cos (t sin (t)) + 10) \cdot (sin (t) + t cos (t))

x \cdot (x - \frac{1}{2}) \cdot (x - 1)

\frac{2 x ^{2} - 2 x + 2}{( 2 x - 1 ) ^{2}}

(2 - 1) (25 + 3)

\frac{2 x ^{3} + 3 x ^{2} + x - 1}{( x + 1 ) ( x ^{2} + 2 x + 2 ) ^{2}}

What is expand (x+Δx)^6 ?
The solution to expand (x+Δx)^6 is x^6+6Δx^6+15Δ^2x^6+20Δ^3x^6+15Δ^4x^6+6Δ^5x^6+Δ^6x^6