What is expand (1-x^2)^5 ?

The solution to expand (1-x^2)^5 is 1-5x^2+10x^4-10x^6+5x^8-x^{10}

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expand (1-x^2)^5

expand

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

1 - 5 x^{2} + 10 x^{4} - 10 x^{6} + 5 x^{8} - x^{10}

Solution steps

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

Apply binomial theorem:

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

a = 1, b = - x^{2}

= i = 0 \sum 5 (i 5) \cdot 1^{(5 - i)} (- x^{2})^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (- x^{2})^{0} : 1

Simplify

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

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (- x^{2})^{2} : 10 x^{4}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (- x^{2})^{3} : - 10 x^{6}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (- x^{2})^{4} : 5 x^{8}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (- x^{2})^{5} : - x^{10}

= 1 - 5 x^{2} + 10 x^{4} - 10 x^{6} + 5 x^{8} - x^{10}

(3 x^{2} - 6 x + 11) - (10 x^{2} - 4 x + 6)

\frac{24 z + 13}{z ^{2} - 100} - \frac{23 z + 3}{z ^{2} - 100}

\frac{9}{2} + 6

- 12 \cdot (- 4)

m^{2} \cdot m^{2}

What is expand (1-x^2)^5 ?
The solution to expand (1-x^2)^5 is 1-5x^2+10x^4-10x^6+5x^8-x^{10}