Solve -42/left(3x-6right)^3+78/left(3x-15right)^3

$$\frac{ -42 }{ { \left(3x-6 \right) }^{ 3 } } + \frac{ 78 }{ { \left(3x-15 \right) }^{ 3 } }$$

Evaluate

$$\frac{2\left(2x^{3}+9x^{2}-123x+257\right)}{3\left(\left(x-5\right)\left(x-2\right)\right)^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(3x-6\right)^{3} and \left(3x-15\right)^{3} is 27\left(x-5\right)^{3}\left(x-2\right)^{3}. Multiply \frac{-42}{\left(3x-6\right)^{3}} times \frac{\left(x-5\right)^{3}}{\left(x-5\right)^{3}}. Multiply \frac{78}{\left(3x-15\right)^{3}} times \frac{\left(x-2\right)^{3}}{\left(x-2\right)^{3}}.

$$\frac{-42\left(x-5\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}+\frac{78\left(x-2\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Since \frac{-42\left(x-5\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}} and \frac{78\left(x-2\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}} have the same denominator, add them by adding their numerators.

$$\frac{-42\left(x-5\right)^{3}+78\left(x-2\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Do the multiplications in -42\left(x-5\right)^{3}+78\left(x-2\right)^{3}.

$$\frac{-42x^{3}+630x^{2}-3150x+5250+78x^{3}-468x^{2}+936x-624}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Combine like terms in -42x^{3}+630x^{2}-3150x+5250+78x^{3}-468x^{2}+936x-624.

$$\frac{36x^{3}+162x^{2}-2214x+4626}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Factor the expressions that are not already factored in \frac{36x^{3}+162x^{2}-2214x+4626}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}.

$$\frac{18\left(2x^{3}+9x^{2}-123x+257\right)}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Cancel out 9 in both numerator and denominator.

$$\frac{2\left(2x^{3}+9x^{2}-123x+257\right)}{3\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Expand 3\left(x-5\right)^{3}\left(x-2\right)^{3}.

$$\frac{2\left(2x^{3}+9x^{2}-123x+257\right)}{3x^{6}-63x^{5}+531x^{4}-2289x^{3}+5310x^{2}-6300x+3000}$$

Use the distributive property to multiply 2 by 2x^{3}+9x^{2}-123x+257.

$$\frac{4x^{3}+18x^{2}-246x+514}{3x^{6}-63x^{5}+531x^{4}-2289x^{3}+5310x^{2}-6300x+3000}$$

Expand

$$\frac{2\left(2x^{3}+9x^{2}-123x+257\right)}{3\left(\left(x-5\right)\left(x-2\right)\right)^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(3x-6\right)^{3} and \left(3x-15\right)^{3} is 27\left(x-5\right)^{3}\left(x-2\right)^{3}. Multiply \frac{-42}{\left(3x-6\right)^{3}} times \frac{\left(x-5\right)^{3}}{\left(x-5\right)^{3}}. Multiply \frac{78}{\left(3x-15\right)^{3}} times \frac{\left(x-2\right)^{3}}{\left(x-2\right)^{3}}.

$$\frac{-42\left(x-5\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}+\frac{78\left(x-2\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Since \frac{-42\left(x-5\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}} and \frac{78\left(x-2\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}} have the same denominator, add them by adding their numerators.

$$\frac{-42\left(x-5\right)^{3}+78\left(x-2\right)^{3}}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Do the multiplications in -42\left(x-5\right)^{3}+78\left(x-2\right)^{3}.

$$\frac{-42x^{3}+630x^{2}-3150x+5250+78x^{3}-468x^{2}+936x-624}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Combine like terms in -42x^{3}+630x^{2}-3150x+5250+78x^{3}-468x^{2}+936x-624.

$$\frac{36x^{3}+162x^{2}-2214x+4626}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Factor the expressions that are not already factored in \frac{36x^{3}+162x^{2}-2214x+4626}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}.

$$\frac{18\left(2x^{3}+9x^{2}-123x+257\right)}{27\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Cancel out 9 in both numerator and denominator.

$$\frac{2\left(2x^{3}+9x^{2}-123x+257\right)}{3\left(x-5\right)^{3}\left(x-2\right)^{3}}$$

Expand 3\left(x-5\right)^{3}\left(x-2\right)^{3}.

$$\frac{2\left(2x^{3}+9x^{2}-123x+257\right)}{3x^{6}-63x^{5}+531x^{4}-2289x^{3}+5310x^{2}-6300x+3000}$$

Use the distributive property to multiply 2 by 2x^{3}+9x^{2}-123x+257.

$$\frac{4x^{3}+18x^{2}-246x+514}{3x^{6}-63x^{5}+531x^{4}-2289x^{3}+5310x^{2}-6300x+3000}$$