Solve left(x^2/2-3x+1/3right)left(6x^2-6right)+(x^3-11x+31)

$$\left( \frac{ { x }^{ 2 } }{ 2 } -3x+ \frac{ 1 }{ 3 } \right) \left( 6 { x }^{ 2 } -6 \right) +( { x }^{ 3 } -11x+31)$$

Evaluate

$$3x^{4}-17x^{3}-x^{2}+7x+29$$

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{x^{2}}{2} times \frac{3}{3}. Multiply \frac{1}{3} times \frac{2}{2}.

$$\left(\frac{3x^{2}}{6}-3x+\frac{2}{6}\right)\left(6x^{2}-6\right)+x^{3}-11x+31$$

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

$$\left(\frac{3x^{2}+2}{6}-3x\right)\left(6x^{2}-6\right)+x^{3}-11x+31$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3x times \frac{6}{6}.

$$\left(\frac{3x^{2}+2}{6}+\frac{6\left(-3\right)x}{6}\right)\left(6x^{2}-6\right)+x^{3}-11x+31$$

Since \frac{3x^{2}+2}{6} and \frac{6\left(-3\right)x}{6} have the same denominator, add them by adding their numerators.

$$\frac{3x^{2}+2+6\left(-3\right)x}{6}\left(6x^{2}-6\right)+x^{3}-11x+31$$

Do the multiplications in 3x^{2}+2+6\left(-3\right)x.

$$\frac{3x^{2}+2-18x}{6}\left(6x^{2}-6\right)+x^{3}-11x+31$$

Express \frac{3x^{2}+2-18x}{6}\left(6x^{2}-6\right) as a single fraction.

$$\frac{\left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)}{6}+x^{3}-11x+31$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{3}-11x+31 times \frac{6}{6}.

$$\frac{\left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)}{6}+\frac{6\left(x^{3}-11x+31\right)}{6}$$

Since \frac{\left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)}{6} and \frac{6\left(x^{3}-11x+31\right)}{6} have the same denominator, add them by adding their numerators.

$$\frac{\left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)+6\left(x^{3}-11x+31\right)}{6}$$

Do the multiplications in \left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)+6\left(x^{3}-11x+31\right).

$$\frac{18x^{4}-18x^{2}+12x^{2}-12-108x^{3}+108x+6x^{3}-66x+186}{6}$$

Combine like terms in 18x^{4}-18x^{2}+12x^{2}-12-108x^{3}+108x+6x^{3}-66x+186.

$$\frac{18x^{4}-6x^{2}+174-102x^{3}+42x}{6}$$

Divide each term of 18x^{4}-6x^{2}+174-102x^{3}+42x by 6 to get 3x^{4}-x^{2}+29-17x^{3}+7x.

$$3x^{4}-x^{2}+29-17x^{3}+7x$$

Expand

$$3x^{4}-17x^{3}-x^{2}+7x+29$$

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{x^{2}}{2} times \frac{3}{3}. Multiply \frac{1}{3} times \frac{2}{2}.

$$\left(\frac{3x^{2}}{6}-3x+\frac{2}{6}\right)\left(6x^{2}-6\right)+x^{3}-11x+31$$

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

$$\left(\frac{3x^{2}+2}{6}-3x\right)\left(6x^{2}-6\right)+x^{3}-11x+31$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3x times \frac{6}{6}.

$$\left(\frac{3x^{2}+2}{6}+\frac{6\left(-3\right)x}{6}\right)\left(6x^{2}-6\right)+x^{3}-11x+31$$

Since \frac{3x^{2}+2}{6} and \frac{6\left(-3\right)x}{6} have the same denominator, add them by adding their numerators.

$$\frac{3x^{2}+2+6\left(-3\right)x}{6}\left(6x^{2}-6\right)+x^{3}-11x+31$$

Do the multiplications in 3x^{2}+2+6\left(-3\right)x.

$$\frac{3x^{2}+2-18x}{6}\left(6x^{2}-6\right)+x^{3}-11x+31$$

Express \frac{3x^{2}+2-18x}{6}\left(6x^{2}-6\right) as a single fraction.

$$\frac{\left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)}{6}+x^{3}-11x+31$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{3}-11x+31 times \frac{6}{6}.

$$\frac{\left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)}{6}+\frac{6\left(x^{3}-11x+31\right)}{6}$$

Since \frac{\left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)}{6} and \frac{6\left(x^{3}-11x+31\right)}{6} have the same denominator, add them by adding their numerators.

$$\frac{\left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)+6\left(x^{3}-11x+31\right)}{6}$$

Do the multiplications in \left(3x^{2}+2-18x\right)\left(6x^{2}-6\right)+6\left(x^{3}-11x+31\right).

$$\frac{18x^{4}-18x^{2}+12x^{2}-12-108x^{3}+108x+6x^{3}-66x+186}{6}$$

Combine like terms in 18x^{4}-18x^{2}+12x^{2}-12-108x^{3}+108x+6x^{3}-66x+186.

$$\frac{18x^{4}-6x^{2}+174-102x^{3}+42x}{6}$$

Divide each term of 18x^{4}-6x^{2}+174-102x^{3}+42x by 6 to get 3x^{4}-x^{2}+29-17x^{3}+7x.

$$3x^{4}-x^{2}+29-17x^{3}+7x$$