What is expand (2-x)^5 ?

The solution to expand (2-x)^5 is 32-80x+80x^2-40x^3+10x^4-x^5

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expand (2-x)^5

expand

(2 - x)^{5}

32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} - x^{5}

Solution steps

(2 - x)^{5}

Apply binomial theorem:

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

a = 2, b = - x

= i = 0 \sum 5 (i 5) \cdot 2^{(5 - i)} (- x)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 2^{5} (- x)^{0} : 32

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 2^{4} (- x)^{1} : - 80 x

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 2^{3} (- x)^{2} : 80 x^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 2^{2} (- x)^{3} : - 40 x^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 2^{1} (- x)^{4} : 10 x^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 2^{0} (- x)^{5} : - x^{5}

= 32 - 80 x + 80 x^{2} - 40 x^{3} + 10 x^{4} - x^{5}

- 1 \times (- 3)

(2 i) (- 2 i)

(4 x^{5})^{2}

\frac{x - 1}{2 x ^{2} + 4 x + 2} \cdot \frac{( x + 1 ) ^{2}}{x - 1}

(5 x - 9)^{2}

What is expand (2-x)^5 ?
The solution to expand (2-x)^5 is 32-80x+80x^2-40x^3+10x^4-x^5