Solve x(x-5)(x-3)-(x-2)^3

$$x ( x - 5 ) ( x - 3 ) - ( x - 2 ) ^ { 3 }$$

Evaluate

$$8+3x-2x^{2}$$

Solution Steps

Use the distributive property to multiply x by x-5.

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

Use the distributive property to multiply x^{2}-5x by x-3 and combine like terms.

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

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

$$x^{3}-8x^{2}+15x-\left(x^{3}-6x^{2}+12x-8\right)$$

To find the opposite of x^{3}-6x^{2}+12x-8, find the opposite of each term.

$$x^{3}-8x^{2}+15x-x^{3}+6x^{2}-12x+8$$

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

$$-8x^{2}+15x+6x^{2}-12x+8$$

Combine -8x^{2} and 6x^{2} to get -2x^{2}.

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

Combine 15x and -12x to get 3x.

$$-2x^{2}+3x+8$$

Expand

$$8+3x-2x^{2}$$

Solution Steps

Use the distributive property to multiply x by x-5.

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

Use the distributive property to multiply x^{2}-5x by x-3 and combine like terms.

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

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

$$x^{3}-8x^{2}+15x-\left(x^{3}-6x^{2}+12x-8\right)$$

To find the opposite of x^{3}-6x^{2}+12x-8, find the opposite of each term.

$$x^{3}-8x^{2}+15x-x^{3}+6x^{2}-12x+8$$

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

$$-8x^{2}+15x+6x^{2}-12x+8$$

Combine -8x^{2} and 6x^{2} to get -2x^{2}.

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

Combine 15x and -12x to get 3x.

$$-2x^{2}+3x+8$$