Solve left(x-2right)^3+left(x-3right)^3

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

Evaluate

$$\left(2x-5\right)\left(x^{2}-5x+7\right)$$

Solution Steps

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

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

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

$$x^{3}-6x^{2}+12x-8+x^{3}-9x^{2}+27x-27$$

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

$$2x^{3}-6x^{2}+12x-8-9x^{2}+27x-27$$

Combine -6x^{2} and -9x^{2} to get -15x^{2}.

$$2x^{3}-15x^{2}+12x-8+27x-27$$

Combine 12x and 27x to get 39x.

$$2x^{3}-15x^{2}+39x-8-27$$

Subtract 27 from -8 to get -35.

$$2x^{3}-15x^{2}+39x-35$$

Expand

$$2x^{3}-15x^{2}+39x-35$$

Solution Steps

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

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

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

$$x^{3}-6x^{2}+12x-8+x^{3}-9x^{2}+27x-27$$

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

$$2x^{3}-6x^{2}+12x-8-9x^{2}+27x-27$$

Combine -6x^{2} and -9x^{2} to get -15x^{2}.

$$2x^{3}-15x^{2}+12x-8+27x-27$$

Combine 12x and 27x to get 39x.

$$2x^{3}-15x^{2}+39x-8-27$$

Subtract 27 from -8 to get -35.

$$2x^{3}-15x^{2}+39x-35$$