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

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

Evaluate

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

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

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

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

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

Add -\frac{1}{2} and 3 to get \frac{5}{2}.

$$\frac{1}{2}x^{3}-\frac{3}{2}x^{2}+\frac{3}{2}x+\frac{5}{2}$$

Expand

$$\frac{x^{3}-3x^{2}+3x+5}{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

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

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

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

Add -\frac{1}{2} and 3 to get \frac{5}{2}.

$$\frac{1}{2}x^{3}-\frac{3}{2}x^{2}+\frac{3}{2}x+\frac{5}{2}$$