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

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

Solve for x

$$x=2$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

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

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-2\right)^{3}.

$$x^{3}-3x^{2}+3x-1+x^{3}-6x^{2}+12x-8+\left(x-3\right)^{3}=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine x^{3} and x^{3} to get 2x^{3}.

$$2x^{3}-3x^{2}+3x-1-6x^{2}+12x-8+\left(x-3\right)^{3}=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -3x^{2} and -6x^{2} to get -9x^{2}.

$$2x^{3}-9x^{2}+3x-1+12x-8+\left(x-3\right)^{3}=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 3x and 12x to get 15x.

$$2x^{3}-9x^{2}+15x-1-8+\left(x-3\right)^{3}=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Subtract 8 from -1 to get -9.

$$2x^{3}-9x^{2}+15x-9+\left(x-3\right)^{3}=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-3\right)^{3}.

$$2x^{3}-9x^{2}+15x-9+x^{3}-9x^{2}+27x-27=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 2x^{3} and x^{3} to get 3x^{3}.

$$3x^{3}-9x^{2}+15x-9-9x^{2}+27x-27=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine -9x^{2} and -9x^{2} to get -18x^{2}.

$$3x^{3}-18x^{2}+15x-9+27x-27=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Combine 15x and 27x to get 42x.

$$3x^{3}-18x^{2}+42x-9-27=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Subtract 27 from -9 to get -36.

$$3x^{3}-18x^{2}+42x-36=3\left(x-1\right)\left(x-2\right)\left(x-3\right)$$

Use the distributive property to multiply 3 by x-1.

$$3x^{3}-18x^{2}+42x-36=\left(3x-3\right)\left(x-2\right)\left(x-3\right)$$

Use the distributive property to multiply 3x-3 by x-2 and combine like terms.

$$3x^{3}-18x^{2}+42x-36=\left(3x^{2}-9x+6\right)\left(x-3\right)$$

Use the distributive property to multiply 3x^{2}-9x+6 by x-3 and combine like terms.

$$3x^{3}-18x^{2}+42x-36=3x^{3}-18x^{2}+33x-18$$

Subtract 3x^{3} from both sides.

$$3x^{3}-18x^{2}+42x-36-3x^{3}=-18x^{2}+33x-18$$

Combine 3x^{3} and -3x^{3} to get 0.

$$-18x^{2}+42x-36=-18x^{2}+33x-18$$

Add 18x^{2} to both sides.

$$-18x^{2}+42x-36+18x^{2}=33x-18$$

Combine -18x^{2} and 18x^{2} to get 0.

$$42x-36=33x-18$$

Subtract 33x from both sides.

$$42x-36-33x=-18$$

Combine 42x and -33x to get 9x.

$$9x-36=-18$$

Add 36 to both sides.

$$9x=-18+36$$

Add -18 and 36 to get 18.

$$9x=18$$

Divide both sides by 9.

$$x=\frac{18}{9}$$

Divide 18 by 9 to get 2.

$$x=2$$