Solve x-x^3/3!+x^5/5!-x^7/7!

$$x - \frac { x ^ { 3 } } { 3 ! } + \frac { x ^ { 5 } } { 5 ! } - \frac { x ^ { 7 } } { 7 ! }$$

Evaluate

$$-\frac{x^{7}}{5040}+\frac{x^{5}}{120}-\frac{x^{3}}{6}+x$$

Short Solution Steps

The factorial of 3 is 6.

$$x-\frac{x^{3}}{6}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}$$

The factorial of 5 is 120.

$$x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{7!}$$

The factorial of 7 is 5040.

$$x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{6}{6}.

$$\frac{6x}{6}-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}$$

Since \frac{6x}{6} and \frac{x^{3}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6x-x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and 120 is 120. Multiply \frac{6x-x^{3}}{6} times \frac{20}{20}.

$$\frac{20\left(6x-x^{3}\right)}{120}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}$$

Since \frac{20\left(6x-x^{3}\right)}{120} and \frac{x^{5}}{120} have the same denominator, add them by adding their numerators.

$$\frac{20\left(6x-x^{3}\right)+x^{5}}{120}-\frac{x^{7}}{5040}$$

Do the multiplications in 20\left(6x-x^{3}\right)+x^{5}.

$$\frac{120x-20x^{3}+x^{5}}{120}-\frac{x^{7}}{5040}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 120 and 5040 is 5040. Multiply \frac{120x-20x^{3}+x^{5}}{120} times \frac{42}{42}.

$$\frac{42\left(120x-20x^{3}+x^{5}\right)}{5040}-\frac{x^{7}}{5040}$$

Since \frac{42\left(120x-20x^{3}+x^{5}\right)}{5040} and \frac{x^{7}}{5040} have the same denominator, subtract them by subtracting their numerators.

$$\frac{42\left(120x-20x^{3}+x^{5}\right)-x^{7}}{5040}$$

Do the multiplications in 42\left(120x-20x^{3}+x^{5}\right)-x^{7}.

$$\frac{5040x-840x^{3}+42x^{5}-x^{7}}{5040}$$

Factor

$$\frac{x\left(5040-840x^{2}+42x^{4}-x^{6}\right)}{5040}$$

Solution Steps

The factorial of 3 is 6.

$$factor(x-\frac{x^{3}}{6}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!})$$

The factorial of 5 is 120.

$$factor(x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{7!})$$

The factorial of 7 is 5040.

$$factor(x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{6}{6}.

$$factor(\frac{6x}{6}-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040})$$

Since \frac{6x}{6} and \frac{x^{3}}{6} have the same denominator, subtract them by subtracting their numerators.

$$factor(\frac{6x-x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and 120 is 120. Multiply \frac{6x-x^{3}}{6} times \frac{20}{20}.

$$factor(\frac{20\left(6x-x^{3}\right)}{120}+\frac{x^{5}}{120}-\frac{x^{7}}{5040})$$

Since \frac{20\left(6x-x^{3}\right)}{120} and \frac{x^{5}}{120} have the same denominator, add them by adding their numerators.

$$factor(\frac{20\left(6x-x^{3}\right)+x^{5}}{120}-\frac{x^{7}}{5040})$$

Do the multiplications in 20\left(6x-x^{3}\right)+x^{5}.

$$factor(\frac{120x-20x^{3}+x^{5}}{120}-\frac{x^{7}}{5040})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 120 and 5040 is 5040. Multiply \frac{120x-20x^{3}+x^{5}}{120} times \frac{42}{42}.

$$factor(\frac{42\left(120x-20x^{3}+x^{5}\right)}{5040}-\frac{x^{7}}{5040})$$

Since \frac{42\left(120x-20x^{3}+x^{5}\right)}{5040} and \frac{x^{7}}{5040} have the same denominator, subtract them by subtracting their numerators.

$$factor(\frac{42\left(120x-20x^{3}+x^{5}\right)-x^{7}}{5040})$$

Do the multiplications in 42\left(120x-20x^{3}+x^{5}\right)-x^{7}.

$$factor(\frac{5040x-840x^{3}+42x^{5}-x^{7}}{5040})$$

Consider 5040x-840x^{3}+42x^{5}-x^{7}. Factor out x.

$$x\left(5040-840x^{2}+42x^{4}-x^{6}\right)$$

Rewrite the complete factored expression. Polynomial 5040-840x^{2}+42x^{4}-x^{6} is not factored since it does not have any rational roots.

$$\frac{x\left(5040-840x^{2}+42x^{4}-x^{6}\right)}{5040}$$