Chapter 12: Q. 30 (page 839)

In Problems 30 and 31, expand each expression using the Binomial Theorem.
${(x + 2)}^{5}$

Short Answer

Expert verified

The expansion is:

${(x + 2)}^{5} = x^{5} + 10 x^{4} + 40 x^{3} + 80 x^{2} + 80 x + 32$

Step by step solution

Step 1. Given information

The expression is:

${(x + 2)}^{5}$

Step 2. Find a and n

By comparing ${(x + 2)}^{5}$ with standard binomial expression, we get

$a = 2 n = 5$

Step 3. Expand the expression.

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

Now

$\begin{array}{rcl} (\begin{matrix} 5 \\ 0 \end{matrix}) & = & \frac{5!}{0! (5 - 0)!} \\ = & \frac{5!}{1.5!} \\ = & 1 \end{array}$

$\begin{array}{rcl} (\begin{matrix} 5 \\ 1 \end{matrix}) & = & \frac{5!}{1! (5 - 1)!} \\ = & \frac{5!}{1.4!} \\ = & \frac{5 \times 4!}{4!} \\ = & 5 \end{array}$

$\begin{array}{rcl} (\begin{matrix} 5 \\ 2 \end{matrix}) & = & \frac{5!}{2! (5 - 2)!} \\ = & \frac{5!}{2! 3!} \\ = & \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 3 \times 2 \times 1} \\ = & 10 \end{array}$

$\begin{array}{rcl} (\begin{matrix} 5 \\ 3 \end{matrix}) & = & \frac{5!}{3! (5 - 3)!} \\ = & \frac{5!}{3! . 2!} \\ = & 10 \end{array}$

$\begin{array}{rcl} (\begin{matrix} 5 \\ 4 \end{matrix}) & = & \frac{5!}{4! (5 - 4)!} \\ = & \frac{5!}{1! . 4!} \\ = & 5 \end{array}$

$\begin{array}{rcl} (\begin{matrix} 5 \\ 5 \end{matrix}) & = & \frac{5!}{5! (5 - 5)!} \\ = & \frac{5!}{1.5!} \\ = & 1 \end{array}$

By substituting these values we get

${(x + 2)}^{5} = 1 . x^{5} + 5 . x^{4} . 2 + 10 . x^{3} . 4 + 10 . x^{2} . 8 + 5 . x . 16 + 1.32 = x^{5} + 10 x^{4} + 40 x^{3} + 80 x^{2} + 80 x + 32$

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In Problems 30 and 31, expand each expression using the Binomial Theorem.
${(x + 2)}^{5}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find a and n

Step 3. Expand the expression.

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Short Answer

Step by step solution

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One App. One Place for Learning.

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In Problems 30 and 31, expand each expression using the Binomial Theorem.
${(x + 2)}^{5}$