Name: Binomial Expansion: Illustration
Uploaded: 2019-07-19
Duration: 4 min 17 s
Description: We already know what Binomial Theorem is. Let's watch this video to observe an illustration that will help us understand Binomial Expansion better.

Question

Find the coefficient of

x^{n}

in the expansion of

(1 + x) (1 - x)^{n}

.

Accepted Answer

Given $(1 + x) (1 - x)^{n}$

$= (1 + x) \times \sum_{r = 0}^{n} (\begin{array}{l} n \\ r \end{array}) 1^{n - r} \times {(- x)}^{r}$

$= (1 + x) \{(\begin{array}{l} n \\ 0 \end{array}) {(- x)}^{0} + (\begin{array}{l} n \\ 1 \end{array}) (- x)^{1} + (\begin{array}{l} n \\ 2 \end{array}) (- x)^{2} + . . + (\begin{matrix} n \\ n - 1 \end{matrix}) (- x)^{n - 1} + (\begin{array}{l} n \\ n \end{array}) (- x)^{n}\}$

$= (1 + x) \{(\begin{array}{l} n \\ 0 \end{array}) + (\begin{array}{l} n \\ 1 \end{array}) (- x)^{1} + (\begin{array}{l} n \\ 2 \end{array}) (x)^{2} + . . + (\begin{matrix} n \\ n - 1 \end{matrix}) {(- 1)}^{n - 1} (x)^{n - 1} + (\begin{array}{l} n \\ n \end{array}) {(- 1)}^{n} (x)^{n}\}$

Thus, the coefficient of $x^{n}$ is, ${(- 1)}^{n} [- (\begin{matrix} n \\ n - 1 \end{matrix}) + (\begin{matrix} n \\ n \end{matrix})] = {(- 1)}^{n} (1 - n)$ .

Find the coefficient of $x^{n}$ in the expansion of $(1 + x) (1 - x)^{n}$ .

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Find the coefficient of xn in the expansion of (1+x)(1-x)n.

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !

Find the coefficient of $x^{n}$ in the expansion of $(1 + x) (1 - x)^{n}$ .