Solve frac{a^{2}+b^{2}+c^2}{1}-a^3+b^3+c^3/3

$$\frac { a ^ { 2 } + b ^ { 2 } + c ^ { 2 } } { 1 } - \frac { a ^ { 3 } + b ^ { 3 } + c ^ { 3 } } { 3 }$$

Evaluate

$$-\frac{a^{3}}{3}-\frac{b^{3}}{3}-\frac{c^{3}}{3}+a^{2}+b^{2}+c^{2}$$

Short Solution Steps

Anything divided by one gives itself.

$$a^{2}+b^{2}+c^{2}-\frac{a^{3}+b^{3}+c^{3}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{2}+b^{2}+c^{2} times \frac{3}{3}.

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

Since \frac{3\left(a^{2}+b^{2}+c^{2}\right)}{3} and \frac{a^{3}+b^{3}+c^{3}}{3} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(a^{2}+b^{2}+c^{2}\right)-\left(a^{3}+b^{3}+c^{3}\right).

$$\frac{3a^{2}+3b^{2}+3c^{2}-a^{3}-b^{3}-c^{3}}{3}$$

Expand

$$-\frac{a^{3}}{3}-\frac{b^{3}}{3}-\frac{c^{3}}{3}+a^{2}+b^{2}+c^{2}$$

Short Solution Steps

Anything divided by one gives itself.

$$a^{2}+b^{2}+c^{2}-\frac{a^{3}+b^{3}+c^{3}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{2}+b^{2}+c^{2} times \frac{3}{3}.

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

Since \frac{3\left(a^{2}+b^{2}+c^{2}\right)}{3} and \frac{a^{3}+b^{3}+c^{3}}{3} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(a^{2}+b^{2}+c^{2}\right)-\left(a^{3}+b^{3}+c^{3}\right).

$$\frac{3a^{2}+3b^{2}+3c^{2}-a^{3}-b^{3}-c^{3}}{3}$$