Solve f(x)=(2x-x^2)^3

$$f ( x ) = ( 2 x - x ^ { 2 } ) ^ { 3 }$$

Evaluate

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

Solution Steps

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

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

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

$$8x^{3}-12x^{4}+6x\left(x^{2}\right)^{2}-\left(x^{2}\right)^{3}$$

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

$$8x^{3}-12x^{4}+6xx^{4}-\left(x^{2}\right)^{3}$$

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

$$8x^{3}-12x^{4}+6x^{5}-\left(x^{2}\right)^{3}$$

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

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

Expand

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

Solution Steps

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

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

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

$$8x^{3}-12x^{4}+6x\left(x^{2}\right)^{2}-\left(x^{2}\right)^{3}$$

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

$$8x^{3}-12x^{4}+6xx^{4}-\left(x^{2}\right)^{3}$$

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

$$8x^{3}-12x^{4}+6x^{5}-\left(x^{2}\right)^{3}$$

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

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