Solve x/1-x^3/3!+x^5/5!-x^7/7!+frac{x^9}{9!}

$$\frac { x } { 1 } - \frac { x ^ { 3 } } { 3 ! } + \frac { x ^ { 5 } } { 5 ! } - \frac { x ^ { 7 } } { 7 ! } + \frac { x ^ { 9 } } { 9 ! }$$

Evaluate

$$\frac{x^{9}}{362880}-\frac{x^{7}}{5040}+\frac{x^{5}}{120}-\frac{x^{3}}{6}+x$$

Short Solution Steps

Anything divided by one gives itself.

$$x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}$$

The factorial of 3 is 6.

$$x-\frac{x^{3}}{6}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}$$

The factorial of 5 is 120.

$$x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!}$$

The factorial of 7 is 5040.

$$x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{9!}$$

The factorial of 9 is 362880.

$$x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880}$$

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

$$\frac{6x}{6}-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880}$$

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

$$\frac{6x-x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880}$$

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}.

$$\frac{20\left(6x-x^{3}\right)}{120}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880}$$

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

$$\frac{20\left(6x-x^{3}\right)+x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880}$$

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

$$\frac{120x-20x^{3}+x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880}$$

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}.

$$\frac{42\left(120x-20x^{3}+x^{5}\right)}{5040}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880}$$

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.

$$\frac{42\left(120x-20x^{3}+x^{5}\right)-x^{7}}{5040}+\frac{x^{9}}{362880}$$

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

$$\frac{5040x-840x^{3}+42x^{5}-x^{7}}{5040}+\frac{x^{9}}{362880}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5040 and 362880 is 362880. Multiply \frac{5040x-840x^{3}+42x^{5}-x^{7}}{5040} times \frac{72}{72}.

$$\frac{72\left(5040x-840x^{3}+42x^{5}-x^{7}\right)}{362880}+\frac{x^{9}}{362880}$$

Since \frac{72\left(5040x-840x^{3}+42x^{5}-x^{7}\right)}{362880} and \frac{x^{9}}{362880} have the same denominator, add them by adding their numerators.

$$\frac{72\left(5040x-840x^{3}+42x^{5}-x^{7}\right)+x^{9}}{362880}$$

Do the multiplications in 72\left(5040x-840x^{3}+42x^{5}-x^{7}\right)+x^{9}.

$$\frac{362880x-60480x^{3}+3024x^{5}-72x^{7}+x^{9}}{362880}$$

Factor

$$\frac{x\left(x^{8}-72x^{6}+3024x^{4}-60480x^{2}+362880\right)}{362880}$$

Solution Steps

Anything divided by one gives itself.

$$factor(x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!})$$

The factorial of 3 is 6.

$$factor(x-\frac{x^{3}}{6}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!})$$

The factorial of 5 is 120.

$$factor(x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{7!}+\frac{x^{9}}{9!})$$

The factorial of 7 is 5040.

$$factor(x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{9!})$$

The factorial of 9 is 362880.

$$factor(x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880})$$

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

$$factor(\frac{6x}{6}-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880})$$

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

$$factor(\frac{6x-x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880})$$

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}.

$$factor(\frac{20\left(6x-x^{3}\right)}{120}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880})$$

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

$$factor(\frac{20\left(6x-x^{3}\right)+x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880})$$

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

$$factor(\frac{120x-20x^{3}+x^{5}}{120}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880})$$

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}.

$$factor(\frac{42\left(120x-20x^{3}+x^{5}\right)}{5040}-\frac{x^{7}}{5040}+\frac{x^{9}}{362880})$$

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.

$$factor(\frac{42\left(120x-20x^{3}+x^{5}\right)-x^{7}}{5040}+\frac{x^{9}}{362880})$$

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

$$factor(\frac{5040x-840x^{3}+42x^{5}-x^{7}}{5040}+\frac{x^{9}}{362880})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5040 and 362880 is 362880. Multiply \frac{5040x-840x^{3}+42x^{5}-x^{7}}{5040} times \frac{72}{72}.

$$factor(\frac{72\left(5040x-840x^{3}+42x^{5}-x^{7}\right)}{362880}+\frac{x^{9}}{362880})$$

Since \frac{72\left(5040x-840x^{3}+42x^{5}-x^{7}\right)}{362880} and \frac{x^{9}}{362880} have the same denominator, add them by adding their numerators.

$$factor(\frac{72\left(5040x-840x^{3}+42x^{5}-x^{7}\right)+x^{9}}{362880})$$

Do the multiplications in 72\left(5040x-840x^{3}+42x^{5}-x^{7}\right)+x^{9}.

$$factor(\frac{362880x-60480x^{3}+3024x^{5}-72x^{7}+x^{9}}{362880})$$

Consider 362880x-60480x^{3}+3024x^{5}-72x^{7}+x^{9}. Factor out x.

$$x\left(362880-60480x^{2}+3024x^{4}-72x^{6}+x^{8}\right)$$

Rewrite the complete factored expression. Polynomial 362880-60480x^{2}+3024x^{4}-72x^{6}+x^{8} is not factored since it does not have any rational roots.

$$\frac{x\left(362880-60480x^{2}+3024x^{4}-72x^{6}+x^{8}\right)}{362880}$$