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Popular Algebra >

expand (x+Δx)^{10}

Solution

expand

(x + Δ x)^{10}

Solution

x^{10} + 10Δ x^{10} + 45 Δ^{2} x^{10} + 120 Δ^{3} x^{10} + 210 Δ^{4} x^{10} + 252 Δ^{5} x^{10} + 210 Δ^{6} x^{10} + 120 Δ^{7} x^{10} + 45 Δ^{8} x^{10} + 10 Δ^{9} x^{10} + Δ^{10} x^{10}

Solution steps

(x + Δ x)^{10}

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 10 (i 10) x^{(10 - i)} (Δ x)^{i}

Expand summation

= \frac{10 !}{0 ! ( 10 - 0 ) !} x^{10} (Δ x)^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} x^{9} (Δ x)^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} x^{8} (Δ x)^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} x^{7} (Δ x)^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} x^{6} (Δ x)^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} x^{5} (Δ x)^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} x^{4} (Δ x)^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} x^{3} (Δ x)^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} x^{2} (Δ x)^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} x^{1} (Δ x)^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} x^{0} (Δ x)^{10}

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} x^{10} (Δ x)^{0} : x^{10}

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} x^{9} (Δ x)^{1} : 10Δ x^{10}

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} x^{8} (Δ x)^{2} : 45 Δ^{2} x^{10}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} x^{7} (Δ x)^{3} : 120 Δ^{3} x^{10}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} x^{6} (Δ x)^{4} : 210 Δ^{4} x^{10}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} x^{5} (Δ x)^{5} : 252 Δ^{5} x^{10}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} x^{4} (Δ x)^{6} : 210 Δ^{6} x^{10}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} x^{3} (Δ x)^{7} : 120 Δ^{7} x^{10}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} x^{2} (Δ x)^{8} : 45 Δ^{8} x^{10}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} x^{1} (Δ x)^{9} : 10 Δ^{9} x^{10}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} x^{0} (Δ x)^{10} : Δ^{10} x^{10}

= x^{10} + 10Δ x^{10} + 45 Δ^{2} x^{10} + 120 Δ^{3} x^{10} + 210 Δ^{4} x^{10} + 252 Δ^{5} x^{10} + 210 Δ^{6} x^{10} + 120 Δ^{7} x^{10} + 45 Δ^{8} x^{10} + 10 Δ^{9} x^{10} + Δ^{10} x^{10}

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