What is expand (x+1)^6 ?

The solution to expand (x+1)^6 is x^6+6x^5+15x^4+20x^3+15x^2+6x+1

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expand (x+1)^6

expand

(x + 1)^{6}

x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1

Solution steps

(x + 1)^{6}

Apply binomial theorem:

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

a = x, b = 1

= i = 0 \sum 6 (i 6) x^{(6 - i)} \cdot 1^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} \cdot 1^{0} : x^{6}

Simplify

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

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} \cdot 1^{2} : 15 x^{4}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} \cdot 1^{3} : 20 x^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} \cdot 1^{4} : 15 x^{2}

Simplify

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

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} \cdot 1^{6} : 1

= x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1

\frac{2}{3 x ^{2} + 12 x} + \frac{8}{2 x}

(x + 1)^{2} - 2

(12 + 5 i) - (5 - 8 i)

4 x^{2} + 2

(- 1 + 2 i) - (12 + 7 i)

What is expand (x+1)^6 ?
The solution to expand (x+1)^6 is x^6+6x^5+15x^4+20x^3+15x^2+6x+1