Expand each of the expressions in Exercises 1 to 5. (x + 1/x)⁶

Question 5 of 20beginner🔧 ApplyLong Answer4 marks

Expand each of the expressions in Exercises 1 to 5.

\left(x + \frac{1}{x}\right)^6

Correct Answer

The expansion of $\left(x + \frac{1}{x}\right)^6$ is:

x^6 + 6x^4 + 15x^2 + 20 + \frac{15}{x^2} + \frac{6}{x^4} + \frac{1}{x^6}

Exercise: EXERCISE 7.1 | Q: 5 | (Page 8)

For More Understanding

Explanation

This question is from Exercise 7.1, Question 5. Students need to apply the binomial theorem formula:

(a+b)^n = {}^nC_0 \cdot a^n + {}^nC_1 \cdot a^{n-1} \cdot b + \dots + {}^nC_n \cdot b^n

Here $a = x$ , $b = \frac{1}{x}$ , and $n = 6$ . The key steps involve calculating each binomial coefficient ( ${}^6C_0$ through ${}^6C_6$ ), then simplifying each term by combining powers of $x$ . The middle term simplifies to a constant since $x^3$ and $\frac{1}{x^3}$ cancel out.

Solution Steps

Step 1: Apply the binomial theorem: (x + 1/x)^6 = ^6C_0· $x^{6}$ + ^6C_1· $x^{5}$ ·(1/x) + ^6C_2· $x^{4}$ ·(1/x)^2 + ^6C_3· $x^{3}$ ·(1/x)^3 + ^6C_4· $x^{2}$ ·(1/x)^4 + ^6C_5·x·(1/x)^5 + ^6C_6·(1/x)^6
Step 2: Calculate binomial coefficients: ^6C_0 = 1, ^6C_1 = 6, ^6C_2 = 15, ^6C_3 = 20, ^6C_4 = 15, ^6C_5 = 6, ^6C_6 = 1
Step 3: Simplify each term by combining powers of x:
= 1· $x^{6}$ + 6· $x^{4}$ + 15· $x^{2}$ + 20·1 + 15·(1/ $x^{2}$ ) + 6·(1/ $x^{4}$ ) + 1·(1/ $x^{6}$ )
Step 4: Write final expansion: $x^{6}$ + $6x^{4}$ + $15x^{2}$ + 20 + 15/ $x^{2}$ + 6/ $x^{4}$ + 1/ $x^{6}$

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