Solve x^{6}-2x^5+x^4-x*(-x^4-x^3+x+2)

$$x ^ { 6 } - 2 x ^ { 5 } + x ^ { 4 } - x \cdot ( - x ^ { 4 } - x ^ { 3 } + x + 2 )$$

Evaluate

$$x\left(x^{5}-x^{4}+2x^{3}-x-2\right)$$

Solution Steps

Use the distributive property to multiply x by -x^{4}-x^{3}+x+2.

$$x^{6}-2x^{5}+x^{4}-\left(x\left(-x^{4}\right)-x^{4}+x^{2}+2x\right)$$

To find the opposite of x\left(-x^{4}\right)-x^{4}+x^{2}+2x, find the opposite of each term.

$$x^{6}-2x^{5}+x^{4}-x\left(-x^{4}\right)+x^{4}-x^{2}-2x$$

Multiply -1 and -1 to get 1.

$$x^{6}-2x^{5}+x^{4}+xx^{4}+x^{4}-x^{2}-2x$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$x^{6}-2x^{5}+x^{4}+x^{5}+x^{4}-x^{2}-2x$$

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

$$x^{6}-x^{5}+x^{4}+x^{4}-x^{2}-2x$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$x^{6}-x^{5}+2x^{4}-x^{2}-2x$$

Expand

$$x^{6}-x^{5}+2x^{4}-x^{2}-2x$$

Solution Steps

Use the distributive property to multiply x by -x^{4}-x^{3}+x+2.

$$x^{6}-2x^{5}+x^{4}-\left(x\left(-x^{4}\right)-x^{4}+x^{2}+2x\right)$$

To find the opposite of x\left(-x^{4}\right)-x^{4}+x^{2}+2x, find the opposite of each term.

$$x^{6}-2x^{5}+x^{4}-x\left(-x^{4}\right)+x^{4}-x^{2}-2x$$

Multiply -1 and -1 to get 1.

$$x^{6}-2x^{5}+x^{4}+xx^{4}+x^{4}-x^{2}-2x$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$x^{6}-2x^{5}+x^{4}+x^{5}+x^{4}-x^{2}-2x$$

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

$$x^{6}-x^{5}+x^{4}+x^{4}-x^{2}-2x$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$x^{6}-x^{5}+2x^{4}-x^{2}-2x$$