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

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

Evaluate

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

Solution Steps

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{6}-6x^{4}x+12x^{2}x^{2}-8x^{3}+1$$

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

$$x^{6}-6x^{5}+12x^{2}x^{2}-8x^{3}+1$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$x^{6}-6x^{5}+12x^{4}-8x^{3}+1$$

Expand

$$x^{6}-6x^{5}+12x^{4}-8x^{3}+1$$

Solution Steps

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{6}-6x^{4}x+12x^{2}x^{2}-8x^{3}+1$$

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

$$x^{6}-6x^{5}+12x^{2}x^{2}-8x^{3}+1$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$x^{6}-6x^{5}+12x^{4}-8x^{3}+1$$