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Popular >

expand (2-sqrt(1-x))^6

Solution

expand

(2 - 1 - x)^{6}

Solution

- x^{3} - 364 - x + 1 + 184 x - x + 1 - 12 x^{2} 1 - x + 63 x^{2} - 363 x + 365

Solution steps

(2 - 1 - x)^{6}

Apply binomial theorem:

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

a = 2, b = - 1 - x

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 2^{0} (- 1 - x)^{6} : (1 - x)^{3}

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

Expand

240 (- x + 1) : - 240 x + 240

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

Expand

60 (- x + 1)^{2} : 60 x^{2} - 120 x + 60

= 64 - 192 1 - x - 240 x + 240 - 160 (- x + 1)^{3} + 60 x^{2} - 120 x + 60 - 12 (1 - x)^{5} + (1 - x)^{3}

Expand

(1 - x)^{3} : 1 - 3 x + 3 x^{2} - x^{3}

= 64 - 192 1 - x - 240 x + 240 - 160 (- x + 1)^{3} + 60 x^{2} - 120 x + 60 - 12 (1 - x)^{5} + 1 - 3 x + 3 x^{2} - x^{3}

Group like terms

= - 12 (1 - x)^{5} - 160 (- x + 1)^{3} - x^{3} + 60 x^{2} + 3 x^{2} - 240 x - 120 x - 3 x - 192 1 - x + 64 + 240 + 60 + 1

Add similar elements:

60 x^{2} + 3 x^{2} = 63 x^{2}

= - 12 (1 - x)^{5} - 160 (- x + 1)^{3} - x^{3} + 63 x^{2} - 240 x - 120 x - 3 x - 192 1 - x + 64 + 240 + 60 + 1

Add similar elements:

- 240 x - 120 x - 3 x = - 363 x

= - 12 (1 - x)^{5} - 160 (- x + 1)^{3} - x^{3} + 63 x^{2} - 363 x - 192 1 - x + 64 + 240 + 60 + 1

Add the numbers:

64 + 240 + 60 + 1 = 365

= - 12 (1 - x)^{5} - 160 (- x + 1)^{3} - x^{3} + 63 x^{2} - 363 x - 192 1 - x + 365

(1 - x)^{5} = (1 - x)^{2} 1 - x

(- x + 1)^{3} = - x + 1 (- x + 1)

= - 12 (1 - x)^{2} 1 - x - 160 - x + 1 (- x + 1) - x^{3} + 63 x^{2} - 363 x - 192 1 - x + 365

Expand

- 12 (1 - x)^{2} 1 - x : - 12 1 - x + 24 x 1 - x - 12 x^{2} 1 - x

= - 12 1 - x + 24 x 1 - x - 12 x^{2} 1 - x - 160 - x + 1 (- x + 1) - x^{3} + 63 x^{2} - 363 x - 192 1 - x + 365

Expand

- 160 - x + 1 (- x + 1) : 160 x - x + 1 - 160 - x + 1

= - 12 1 - x + 24 x 1 - x - 12 x^{2} 1 - x + 160 x - x + 1 - 160 - x + 1 - x^{3} + 63 x^{2} - 363 x - 192 1 - x + 365

Group like terms

= - x^{3} - 12 1 - x - 160 - x + 1 - 192 1 - x + 24 x 1 - x + 160 x - x + 1 - 12 x^{2} 1 - x + 63 x^{2} - 363 x + 365

Add similar elements:

- 12 1 - x - 160 - x + 1 - 192 1 - x = - 364 - x + 1

= - x^{3} - 364 - x + 1 + 24 x 1 - x + 160 x - x + 1 - 12 x^{2} 1 - x + 63 x^{2} - 363 x + 365

Add similar elements:

24 x 1 - x + 160 x - x + 1 = 184 x - x + 1

= - x^{3} - 364 - x + 1 + 184 x - x + 1 - 12 x^{2} 1 - x + 63 x^{2} - 363 x + 365

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