What is expand (2x-1)^6 ?

The solution to expand (2x-1)^6 is 64x^6-192x^5+240x^4-160x^3+60x^2-12x+1

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expand (2x-1)^6

expand

(2 x - 1)^{6}

64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} - 12 x + 1

Solution steps

(2 x - 1)^{6}

Apply binomial theorem:

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

a = 2 x, b = - 1

= i = 0 \sum 6 (i 6) (2 x)^{(6 - i)} (- 1)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (2 x)^{6} (- 1)^{0} : 64 x^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (2 x)^{5} (- 1)^{1} : - 192 x^{5}

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (2 x)^{4} (- 1)^{2} : 240 x^{4}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (2 x)^{3} (- 1)^{3} : - 160 x^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (2 x)^{2} (- 1)^{4} : 60 x^{2}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (2 x)^{1} (- 1)^{5} : - 12 x

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (2 x)^{0} (- 1)^{6} : 1

= 64 x^{6} - 192 x^{5} + 240 x^{4} - 160 x^{3} + 60 x^{2} - 12 x + 1

ab x^{2} + (a^{2} + b^{2}) x + ab

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

- 6^{4}

\frac{a ^{m + 3}}{a ^{m + 2}}

What is expand (2x-1)^6 ?
The solution to expand (2x-1)^6 is 64x^6-192x^5+240x^4-160x^3+60x^2-12x+1