Solve -23/4a^{5}-(-3/5a^3)-(-3/4a^5)-2/15a^3+frac{1}{5}a^3

$$- \frac { 23 } { 4 } a ^ { 5 } - ( - \frac { 3 } { 5 } a ^ { 3 } ) - ( - \frac { 3 } { 4 } a ^ { 5 } ) - \frac { 2 } { 15 } a ^ { 3 } + \frac { 1 } { 5 } a ^ { 3 }$$

Evaluate

$$-5a^{5}+\frac{2a^{3}}{3}$$

Solution Steps

The opposite of -\frac{3}{5}a^{3} is \frac{3}{5}a^{3}.

$$-\frac{23}{4}a^{5}+\frac{3}{5}a^{3}-\left(-\frac{3}{4}a^{5}\right)-\frac{2}{15}a^{3}+\frac{1}{5}a^{3}$$

The opposite of -\frac{3}{4}a^{5} is \frac{3}{4}a^{5}.

$$-\frac{23}{4}a^{5}+\frac{3}{5}a^{3}+\frac{3}{4}a^{5}-\frac{2}{15}a^{3}+\frac{1}{5}a^{3}$$

Combine -\frac{23}{4}a^{5} and \frac{3}{4}a^{5} to get -5a^{5}.

$$-5a^{5}+\frac{3}{5}a^{3}-\frac{2}{15}a^{3}+\frac{1}{5}a^{3}$$

Combine \frac{3}{5}a^{3} and -\frac{2}{15}a^{3} to get \frac{7}{15}a^{3}.

$$-5a^{5}+\frac{7}{15}a^{3}+\frac{1}{5}a^{3}$$

Combine \frac{7}{15}a^{3} and \frac{1}{5}a^{3} to get \frac{2}{3}a^{3}.

$$-5a^{5}+\frac{2}{3}a^{3}$$

Factor

$$\frac{\left(2-15a^{2}\right)a^{3}}{3}$$

Solution Steps

Factor out \frac{1}{60}.

$$\frac{-345a^{5}+36a^{3}+45a^{5}-8a^{3}+12a^{3}}{60}$$

Consider -345a^{5}+36a^{3}+45a^{5}-8a^{3}+12a^{3}. Factor out a^{3}.

$$a^{3}\left(-300a^{2}+40\right)$$

Consider -345a^{2}+36+45a^{2}-8+12. Multiply and combine like terms.

$$-300a^{2}+40$$

Consider -300a^{2}+40. Factor out 20.

$$20\left(-15a^{2}+2\right)$$

Rewrite the complete factored expression.

$$\frac{a^{3}\left(-15a^{2}+2\right)}{3}$$

Simplify. Polynomial -15a^{2}+2 is not factored since it does not have any rational roots.

$$\frac{\left(-15a^{2}+2\right)a^{3}}{3}$$