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

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

Evaluate

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

Solution Steps

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

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

Use the distributive property to multiply -x by x^{3}-6x^{2}+12x-8.

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

Multiply -6 and -1 to get 6.

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Multiply -8 and -1 to get 8.

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

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$-x^{4}+6x^{3}+12\left(-1\right)xx+8x$$

Multiply x and x to get x^{2}.

$$-x^{4}+6x^{3}+12\left(-1\right)x^{2}+8x$$

Multiply 12 and -1 to get -12.

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

Expand

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

Solution Steps

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

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

Use the distributive property to multiply -x by x^{3}-6x^{2}+12x-8.

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

Multiply -6 and -1 to get 6.

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Multiply -8 and -1 to get 8.

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

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$-x^{4}+6x^{3}+12\left(-1\right)xx+8x$$

Multiply x and x to get x^{2}.

$$-x^{4}+6x^{3}+12\left(-1\right)x^{2}+8x$$

Multiply 12 and -1 to get -12.

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