Solve f(x)=1/2x(x-3)^3

$$f ( x ) = \frac { 1 } { 2 } x ( x - 3 ) ^ { 3 }$$

Evaluate

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

Solution Steps

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

$$\frac{1}{2}x\left(x^{3}-9x^{2}+27x-27\right)$$

Use the distributive property to multiply \frac{1}{2}x by x^{3}-9x^{2}+27x-27.

$$\frac{1}{2}x^{4}-\frac{9}{2}x^{3}+\frac{27}{2}x^{2}-\frac{27}{2}x$$

Expand

$$\frac{x^{4}-9x^{3}+27x^{2}-27x}{2}$$

Solution Steps

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

$$\frac{1}{2}x\left(x^{3}-9x^{2}+27x-27\right)$$

Use the distributive property to multiply \frac{1}{2}x by x^{3}-9x^{2}+27x-27.

$$\frac{1}{2}x^{4}-\frac{9}{2}x^{3}+\frac{27}{2}x^{2}-\frac{27}{2}x$$