Solve f(x)=1/27(x+3)^3(x-1)^2(x-4)

$$f ( x ) = \frac { 1 } { 27 } ( x + 3 ) ^ { 3 } ( x - 1 ) ^ { 2 } ( x - 4 )$$

Evaluate

$$\frac{\left(x-4\right)\left(x-1\right)^{2}\left(x+3\right)^{3}}{27}$$

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}.

$$\frac{1}{27}\left(x^{3}+9x^{2}+27x+27\right)\left(x-1\right)^{2}\left(x-4\right)$$

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

$$\frac{1}{27}\left(x^{3}+9x^{2}+27x+27\right)\left(x^{2}-2x+1\right)\left(x-4\right)$$

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

$$\left(\frac{1}{27}x^{3}+\frac{1}{3}x^{2}+x+1\right)\left(x^{2}-2x+1\right)\left(x-4\right)$$

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

$$\left(\frac{1}{27}x^{5}+\frac{7}{27}x^{4}+\frac{10}{27}x^{3}-\frac{2}{3}x^{2}-x+1\right)\left(x-4\right)$$

Use the distributive property to multiply \frac{1}{27}x^{5}+\frac{7}{27}x^{4}+\frac{10}{27}x^{3}-\frac{2}{3}x^{2}-x+1 by x-4 and combine like terms.

$$\frac{1}{27}x^{6}+\frac{1}{9}x^{5}-\frac{2}{3}x^{4}-\frac{58}{27}x^{3}+\frac{5}{3}x^{2}+5x-4$$

Expand

$$\frac{x^{6}}{27}+\frac{x^{5}}{9}-\frac{2x^{4}}{3}-\frac{58x^{3}}{27}+\frac{5x^{2}}{3}+5x-4$$

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}.

$$\frac{1}{27}\left(x^{3}+9x^{2}+27x+27\right)\left(x-1\right)^{2}\left(x-4\right)$$

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

$$\frac{1}{27}\left(x^{3}+9x^{2}+27x+27\right)\left(x^{2}-2x+1\right)\left(x-4\right)$$

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

$$\left(\frac{1}{27}x^{3}+\frac{1}{3}x^{2}+x+1\right)\left(x^{2}-2x+1\right)\left(x-4\right)$$

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

$$\left(\frac{1}{27}x^{5}+\frac{7}{27}x^{4}+\frac{10}{27}x^{3}-\frac{2}{3}x^{2}-x+1\right)\left(x-4\right)$$

Use the distributive property to multiply \frac{1}{27}x^{5}+\frac{7}{27}x^{4}+\frac{10}{27}x^{3}-\frac{2}{3}x^{2}-x+1 by x-4 and combine like terms.

$$\frac{1}{27}x^{6}+\frac{1}{9}x^{5}-\frac{2}{3}x^{4}-\frac{58}{27}x^{3}+\frac{5}{3}x^{2}+5x-4$$