# How do you use the Binomial Theorem to expand  (1/X + X)^6?

Nov 7, 2016

${\left(\frac{1}{x} + x\right)}^{6} = \frac{1}{x} ^ 6 + \frac{6}{x} ^ 4 + \frac{15}{x} ^ 2 + 20 + 15 {x}^{2} + 6 {x}^{4} + \frac{1}{x} ^ 6$

#### Explanation:

Binomial expansion of ${\left(x + a\right)}^{6}$ is

${x}^{6} {+}^{6} {C}_{1} {x}^{5} a {+}^{6} {C}_{2} {x}^{4} {a}^{2} {+}^{6} {C}_{3} {x}^{3} {a}^{3} {+}^{6} {C}_{4} {x}^{2} {a}^{4} {+}^{6} {C}_{5} x {a}^{5} {+}^{6} {C}_{6} {a}^{6}$

And hence binomial expansion of ${\left(\frac{1}{x} + x\right)}^{6}$ or ${\left(x + \frac{1}{x}\right)}^{6}$ is

${x}^{6} {+}^{6} {C}_{1} {x}^{5} \left(\frac{1}{x}\right) {+}^{6} {C}_{2} {x}^{4} {\left(\frac{1}{x}\right)}^{2} {+}^{6} {C}_{3} {x}^{3} {\left(\frac{1}{x}\right)}^{3} {+}^{6} {C}_{4} {x}^{2} {\left(\frac{1}{x}\right)}^{4} {+}^{6} {C}_{5} x {\left(\frac{1}{x}\right)}^{5} {+}^{6} {C}_{6} {\left(\frac{1}{x}\right)}^{6}$

or ${x}^{6} + \frac{6}{1} \times {x}^{4} + \frac{6 \times 5}{1 \times 2} \times {x}^{2} + \frac{6 \times 5 \times 4}{1 \times 2 \times 3} + \frac{6 \times 5 \times 4 \times 3}{1 \times 2 \times 3 \times 4} \times \frac{1}{x} ^ 4 + \frac{6 \times 5 \times 4 \times 3 \times 2}{1 \times 2 \times 3 \times 4 \times 5} \times \frac{1}{x} ^ 6$

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

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