Expand Using Pascal's Triangle (x+1/x)^4

${(x + \frac{1}{x})}^{4}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(x + \frac{1}{x})}^{4}$ , $n = 4$ so the coefficients of the expansion will correspond with line $5$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 4 - 6 - 4 - 1$ .

$1 a^{4} b^{0} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + 1 a^{0} b^{4}$

Step 3

Substitute the actual values of $a$ $x$ and $b$ $\frac{1}{x}$ into the expression.

$1 {(x)}^{4} {(\frac{1}{x})}^{0} + 4 {(x)}^{3} {(\frac{1}{x})}^{1} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Step 4

Simplify each term.

Multiply ${(x)}^{4}$ by $1$ .

${(x)}^{4} {(\frac{1}{x})}^{0} + 4 {(x)}^{3} {(\frac{1}{x})}^{1} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Apply the product rule to $\frac{1}{x}$ .

$x^{4} \frac{1^{0}}{x^{0}} + 4 {(x)}^{3} {(\frac{1}{x})}^{1} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Anything raised to $0$ is $1$ .

$x^{4} \frac{1}{x^{0}} + 4 {(x)}^{3} {(\frac{1}{x})}^{1} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Anything raised to $0$ is $1$ .

$x^{4} \frac{1}{1} + 4 {(x)}^{3} {(\frac{1}{x})}^{1} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Divide $1$ by $1$ .

$x^{4} \cdot 1 + 4 {(x)}^{3} {(\frac{1}{x})}^{1} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Multiply $x^{4}$ by $1$ .

$x^{4} + 4 {(x)}^{3} {(\frac{1}{x})}^{1} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Simplify.

$x^{4} + 4 x^{3} \frac{1}{x} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Cancel the common factor of $x$ .

Factor $x$ out of $4 x^{3}$ .

$x^{4} + x (4 x^{2}) \frac{1}{x} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Cancel the common factor.

$x^{4} + x (4 x^{2}) \frac{1}{x} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Rewrite the expression.

$x^{4} + 4 x^{2} + 6 {(x)}^{2} {(\frac{1}{x})}^{2} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Apply the product rule to $\frac{1}{x}$ .

$x^{4} + 4 x^{2} + 6 x^{2} \frac{1^{2}}{x^{2}} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

One to any power is one.

$x^{4} + 4 x^{2} + 6 x^{2} \frac{1}{x^{2}} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Cancel the common factor of $x^{2}$ .

Factor $x^{2}$ out of $6 x^{2}$ .

$x^{4} + 4 x^{2} + x^{2} \cdot 6 \frac{1}{x^{2}} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Cancel the common factor.

$x^{4} + 4 x^{2} + x^{2} \cdot 6 \frac{1}{x^{2}} + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Rewrite the expression.

$x^{4} + 4 x^{2} + 6 + 4 {(x)}^{1} {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Simplify.

$x^{4} + 4 x^{2} + 6 + 4 x {(\frac{1}{x})}^{3} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Apply the product rule to $\frac{1}{x}$ .

$x^{4} + 4 x^{2} + 6 + 4 x \frac{1^{3}}{x^{3}} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

One to any power is one.

$x^{4} + 4 x^{2} + 6 + 4 x \frac{1}{x^{3}} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Cancel the common factor of $x$ .

Factor $x$ out of $4 x$ .

$x^{4} + 4 x^{2} + 6 + x \cdot 4 \frac{1}{x^{3}} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Factor $x$ out of $x^{3}$ .

$x^{4} + 4 x^{2} + 6 + x \cdot 4 \frac{1}{x \cdot x^{2}} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Cancel the common factor.

$x^{4} + 4 x^{2} + 6 + x \cdot 4 \frac{1}{x \cdot x^{2}} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Rewrite the expression.

$x^{4} + 4 x^{2} + 6 + 4 \frac{1}{x^{2}} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Combine $4$ and $\frac{1}{x^{2}}$ .

$x^{4} + 4 x^{2} + 6 + \frac{4}{x^{2}} + 1 {(x)}^{0} {(\frac{1}{x})}^{4}$

Multiply ${(x)}^{0}$ by $1$ .

$x^{4} + 4 x^{2} + 6 + \frac{4}{x^{2}} + {(x)}^{0} {(\frac{1}{x})}^{4}$

Anything raised to $0$ is $1$ .

$x^{4} + 4 x^{2} + 6 + \frac{4}{x^{2}} + 1 {(\frac{1}{x})}^{4}$

Multiply ${(\frac{1}{x})}^{4}$ by $1$ .

$x^{4} + 4 x^{2} + 6 + \frac{4}{x^{2}} + {(\frac{1}{x})}^{4}$

Apply the product rule to $\frac{1}{x}$ .

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

One to any power is one.

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

Expand Using Pascal's Triangle (x+1/x)^4

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