Solve -x^3/3-x^2/3+(x+1)^3/3+(x+1)^2/3

$$- \frac { x ^ { 3 } } { 3 } - \frac { x ^ { 2 } } { 3 } + \frac { ( x + 1 ) ^ { 3 } } { 3 } + \frac { ( x + 1 ) ^ { 2 } } { 3 }$$

Evaluate

$$x^{2}+\frac{5x}{3}+\frac{2}{3}$$

Short Solution Steps

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

$$\frac{-x^{3}-x^{2}}{3}+\frac{\left(x+1\right)^{3}}{3}+\frac{\left(x+1\right)^{2}}{3}$$

Since \frac{-x^{3}-x^{2}}{3} and \frac{\left(x+1\right)^{3}}{3} have the same denominator, add them by adding their numerators.

$$\frac{-x^{3}-x^{2}+\left(x+1\right)^{3}}{3}+\frac{\left(x+1\right)^{2}}{3}$$

Do the multiplications in -x^{3}-x^{2}+\left(x+1\right)^{3}.

$$\frac{-x^{3}-x^{2}+x^{3}+3x^{2}+3x\times 1^{2}+1^{3}}{3}+\frac{\left(x+1\right)^{2}}{3}$$

Combine like terms in -x^{3}-x^{2}+x^{3}+3x^{2}+3x\times 1^{2}+1^{3}.

$$\frac{2x^{2}+3x+1}{3}+\frac{\left(x+1\right)^{2}}{3}$$

Since \frac{2x^{2}+3x+1}{3} and \frac{\left(x+1\right)^{2}}{3} have the same denominator, add them by adding their numerators.

$$\frac{2x^{2}+3x+1+\left(x+1\right)^{2}}{3}$$

Do the multiplications in 2x^{2}+3x+1+\left(x+1\right)^{2}.

$$\frac{2x^{2}+3x+1+x^{2}+2x+1}{3}$$

Combine like terms in 2x^{2}+3x+1+x^{2}+2x+1.

$$\frac{3x^{2}+5x+2}{3}$$

Expand

$$x^{2}+\frac{5x}{3}+\frac{2}{3}$$

Short Solution Steps

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

$$\frac{-x^{3}-x^{2}}{3}+\frac{\left(x+1\right)^{3}}{3}+\frac{\left(x+1\right)^{2}}{3}$$

Since \frac{-x^{3}-x^{2}}{3} and \frac{\left(x+1\right)^{3}}{3} have the same denominator, add them by adding their numerators.

$$\frac{-x^{3}-x^{2}+\left(x+1\right)^{3}}{3}+\frac{\left(x+1\right)^{2}}{3}$$

Do the multiplications in -x^{3}-x^{2}+\left(x+1\right)^{3}.

$$\frac{-x^{3}-x^{2}+x^{3}+3x^{2}+3x\times 1^{2}+1^{3}}{3}+\frac{\left(x+1\right)^{2}}{3}$$

Combine like terms in -x^{3}-x^{2}+x^{3}+3x^{2}+3x\times 1^{2}+1^{3}.

$$\frac{2x^{2}+3x+1}{3}+\frac{\left(x+1\right)^{2}}{3}$$

Since \frac{2x^{2}+3x+1}{3} and \frac{\left(x+1\right)^{2}}{3} have the same denominator, add them by adding their numerators.

$$\frac{2x^{2}+3x+1+\left(x+1\right)^{2}}{3}$$

Do the multiplications in 2x^{2}+3x+1+\left(x+1\right)^{2}.

$$\frac{2x^{2}+3x+1+x^{2}+2x+1}{3}$$

Combine like terms in 2x^{2}+3x+1+x^{2}+2x+1.

$$\frac{3x^{2}+5x+2}{3}$$