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Chapter 12: Q 26. (page 836)

Expand ${(\sqrt{x} - \sqrt{3})}^{4}$ using Binomial Theorem

Short Answer

Expert verified

The value of ${(\sqrt{x} - \sqrt{3})}^{4} = - 4 \sqrt{3} x^{3 / 2} + x^{2} + 18 x - 12 \sqrt{3} \sqrt{x} + 9$

Step by step solution

01

Step 1. Defination

Let $x$ and $a$ bereal number for any positive integer $n$ , we have ${(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + \dots \dots + (\begin{matrix} n \\ j \end{matrix}) a^{j} x^{n - j} + \dots \dots + (\begin{matrix} n \\ n \end{matrix}) a^{n} = \sum_{j = 0}^{n} (\begin{matrix} n \\ j \end{matrix}) x^{n - j} a^{j}$

02

Step 2. Explanation

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

$\Rightarrow {(\sqrt{x} - \sqrt{3})}^{4} = - 4 \sqrt{3} x^{3 / 2} + x^{2} + 18 x + 2 \sqrt{3} \sqrt{x} + 9$

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Most popular questions from this chapter

In Problems 17–28, write down the first five terms of each sequence.

$\{b_{n}\} = \{\frac{2 n + 1}{2 n}\}$

In Problems 29–36, the given pattern continues. Write down the nth term of a sequence $\{a_{n}\}$ suggested by the pattern.

$1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, 7, \frac{1}{8}, . . .$

Fibonacci Sequence: Let

$u_{n} = \frac{{(1 + \sqrt{5})}^{n} - {(1 - \sqrt{5})}^{n}}{2^{n} \sqrt{5}}$

define the nth term of a sequence.

(a) Show that u₁= 1 and u₂ = 1.

(b) Show that u_n+2 = u_n+1 + u_n.

(c) Draw the conclusion that {u_n} is the Fibonacci sequence.

Education IRA: On January 1, 1999, John’s parents decided to place \(45 at the end of each month into an Education IRA.

(a) Find a recursive formula that represents the balance at the end of each month if the rate of return is assumed to be 6% per annum compounded monthly.

(b) How long will it be before the value of the account exceeds \)4000?

(c) What will be the value of the account in 16 years when John goes to college?

Growth of a Rabbit Colony:

A colony of rabbits begins with one pair of mature rabbits, which will produce a pair of offspring (one male, one female) each month. Assume that all rabbits mature in 1 month and produce a pair of offspring (one male, one female) after 2 months. If no rabbits ever die, how many pairs of mature rabbits are there after 7 months?

[Hint: The Fibonacci sequence models this colony. Do you see why?]

See all solutions

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