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Popular >

expand (z-1)^4

Solution

expand

(z - 1)^{4}

Solution

z^{4} - 4 z^{3} + 6 z^{2} - 4 z + 1

Solution steps

(z - 1)^{4}

Apply binomial theorem:

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

a = z, b = - 1

= i = 0 \sum 4 (i 4) z^{(4 - i)} (- 1)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} z^{4} (- 1)^{0} : z^{4}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} z^{3} (- 1)^{1} : - 4 z^{3}

Simplify

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

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} z^{1} (- 1)^{3} : - 4 z

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} z^{0} (- 1)^{4} : 1

= z^{4} - 4 z^{3} + 6 z^{2} - 4 z + 1

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Frequently Asked Questions (FAQ)

What is expand (z-1)^4 ?
The solution to expand (z-1)^4 is z^4-4z^3+6z^2-4z+1