Solve {l}{-5x^3+5x^2-1+x-1}{-5(x^3-1)+(x-1)(x+1)+(x-1)}

$$\left. \begin{array} { l } { - 5 x ^ { 3 } + 5 x ^ { 2 } - 1 + x - 1 } \\ { - 5 ( x ^ { 3 } - 1 ) + ( x - 1 ) ( x + 1 ) + ( x - 1 ) } \end{array} \right.$$

Evaluate

$$1+2x+6x^{2}-10x^{3}$$

Solution Steps

Subtract 1 from -1 to get -2.

$$-5x^{3}+5x^{2}-2+x-5\left(x^{3}-1\right)+\left(x-1\right)\left(x+1\right)+x-1$$

Use the distributive property to multiply -5 by x^{3}-1.

$$-5x^{3}+5x^{2}-2+x-5x^{3}+5+\left(x-1\right)\left(x+1\right)+x-1$$

Combine -5x^{3} and -5x^{3} to get -10x^{3}.

$$-10x^{3}+5x^{2}-2+x+5+\left(x-1\right)\left(x+1\right)+x-1$$

Add -2 and 5 to get 3.

$$-10x^{3}+5x^{2}+3+x+\left(x-1\right)\left(x+1\right)+x-1$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$-10x^{3}+5x^{2}+3+x+x^{2}-1+x-1$$

Combine 5x^{2} and x^{2} to get 6x^{2}.

$$-10x^{3}+6x^{2}+3+x-1+x-1$$

Subtract 1 from 3 to get 2.

$$-10x^{3}+6x^{2}+2+x+x-1$$

Combine x and x to get 2x.

$$-10x^{3}+6x^{2}+2+2x-1$$

Subtract 1 from 2 to get 1.

$$-10x^{3}+6x^{2}+1+2x$$

Expand

$$1+2x+6x^{2}-10x^{3}$$

Solution Steps

Subtract 1 from -1 to get -2.

$$-5x^{3}+5x^{2}-2+x-5\left(x^{3}-1\right)+\left(x-1\right)\left(x+1\right)+x-1$$

Use the distributive property to multiply -5 by x^{3}-1.

$$-5x^{3}+5x^{2}-2+x-5x^{3}+5+\left(x-1\right)\left(x+1\right)+x-1$$

Combine -5x^{3} and -5x^{3} to get -10x^{3}.

$$-10x^{3}+5x^{2}-2+x+5+\left(x-1\right)\left(x+1\right)+x-1$$

Add -2 and 5 to get 3.

$$-10x^{3}+5x^{2}+3+x+\left(x-1\right)\left(x+1\right)+x-1$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$-10x^{3}+5x^{2}+3+x+x^{2}-1+x-1$$

Combine 5x^{2} and x^{2} to get 6x^{2}.

$$-10x^{3}+6x^{2}+3+x-1+x-1$$

Subtract 1 from 3 to get 2.

$$-10x^{3}+6x^{2}+2+x+x-1$$

Combine x and x to get 2x.

$$-10x^{3}+6x^{2}+2+2x-1$$

Subtract 1 from 2 to get 1.

$$-10x^{3}+6x^{2}+1+2x$$