Solve a^4/2-a^3/3+a^2/2-a

$$\frac { a ^ { 4 } } { 2 } - \frac { a ^ { 3 } } { 3 } + \frac { a ^ { 2 } } { 2 } - a$$

Factor

$$\frac{a\left(3a^{3}-2a^{2}+3a-6\right)}{6}$$

Solution Steps

Factor out \frac{1}{6}.

$$\frac{3a^{4}-2a^{3}+3a^{2}-6a}{6}$$

Consider 3a^{4}-2a^{3}+3a^{2}-6a. Factor out a.

$$a\left(3a^{3}-2a^{2}+3a-6\right)$$

Rewrite the complete factored expression. Polynomial 3a^{3}-2a^{2}+3a-6 is not factored since it does not have any rational roots.

$$\frac{a\left(3a^{3}-2a^{2}+3a-6\right)}{6}$$

Evaluate

$$\frac{a^{4}}{2}-\frac{a^{3}}{3}+\frac{a^{2}}{2}-a$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{a^{4}}{2} times \frac{3}{3}. Multiply \frac{a^{3}}{3} times \frac{2}{2}.

$$\frac{3a^{4}}{6}-\frac{2a^{3}}{6}+\frac{a^{2}}{2}-a$$

Since \frac{3a^{4}}{6} and \frac{2a^{3}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3a^{4}-2a^{3}}{6}+\frac{a^{2}}{2}-a$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and 2 is 6. Multiply \frac{a^{2}}{2} times \frac{3}{3}.

$$\frac{3a^{4}-2a^{3}}{6}+\frac{3a^{2}}{6}-a$$

Since \frac{3a^{4}-2a^{3}}{6} and \frac{3a^{2}}{6} have the same denominator, add them by adding their numerators.

$$\frac{3a^{4}-2a^{3}+3a^{2}}{6}-a$$

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

$$\frac{3a^{4}-2a^{3}+3a^{2}}{6}-\frac{6a}{6}$$

Since \frac{3a^{4}-2a^{3}+3a^{2}}{6} and \frac{6a}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3a^{4}-2a^{3}+3a^{2}-6a}{6}$$