Chapter 11: Q. 12.71. (page 1069)

Use the Binomial Theorem to expand ${(x - 3)}^{5}$

Short Answer

Expert verified

The simplified form of the given expression is $x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243$

Step by step solution

Step 1. Given information.

The given expression is ${(x - 5)}^{5}$

Step 2. Substitute the values and simplify.

Substitute in the values into the binomial expansion,

$a = x b = - 3 n = 5$

${(x - 3)}^{5} = (\begin{matrix} 5 \\ 0 \end{matrix}) x^{5} + (\begin{matrix} 5 \\ 1 \end{matrix}) x^{5 - 1} (- 3) + (\begin{matrix} 5 \\ 2 \end{matrix}) x^{5 - 2} {(- 3)}^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) x^{5 - 3} {(- 3)}^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) x^{5 - 4} {(- 3)}^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) {(- 3)}^{5}$

Simplify the exponents.

${(x - 3)}^{5} = (\begin{matrix} 5 \\ 0 \end{matrix}) x^{5} - (\begin{matrix} 5 \\ 1 \end{matrix}) x^{4} 3 + (\begin{matrix} 5 \\ 2 \end{matrix}) x^{3} 9 - (\begin{matrix} 5 \\ 3 \end{matrix}) x^{2} 27 + (\begin{matrix} 5 \\ 4 \end{matrix}) x 81 - (\begin{matrix} 5 \\ 5 \end{matrix}) 243$

Evaluate the coefficients. Remember,

$(\begin{matrix} n \\ 1 \end{matrix}) = n, (\begin{matrix} n \\ n \end{matrix}) = 1, (\begin{matrix} n \\ 0 \end{matrix}) = 1$

Simplify the coefficients.

${(x - 3)}^{5} = 1 x^{5} - \frac{5!}{1! 4!} x^{4} 3 + \frac{5!}{2! 3!} x^{3} 9 - \frac{5!}{3! 2!} x^{2} 27 + \frac{5!}{4! 1!} x 81 - 243$

Step 3. Conclusion.

Therefore, the expansion of the given expression is ${(x - 3)}^{5} = x^{5} - 15 x^{4} + 90 x^{3} - 270 x^{2} + 405 x - 243$

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Use the Binomial Theorem to expand ${(x - 3)}^{5}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitute the values and simplify.

Step 3. Conclusion.

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Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitute the values and simplify.

Step 3. Conclusion.

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Use the Binomial Theorem to expand ${(x - 3)}^{5}$