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brojevnids4

Answered question

2022-12-14

What is the expansion of

{(x - 1)}^{4}

Answer & Explanation

Aryan Baxter

Beginner2022-12-15Added 3 answers

Determine the expansion of

{(x - 1)}^{4}

.
We are aware that the binomial theorem says

(^{a + b) n} = \sum k = 0 n c k n \cdot (a^{n - k} b^{k})

.
Figure out the expansion of

{(x - 1)}^{4}

:
From the binomial theorem,

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

Hence, the required expansion is

{(x - 1)}^{4} = x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1

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