Solve 1/18left(x+2right)left(x-1right)^2left(x-3right)^2

$$\frac{ 1 }{ 18 } \left( x+2 \right) { \left(x-1 \right) }^{ 2 } { \left(x-3 \right) }^{ 2 }$$

Evaluate

$$\frac{\left(x+2\right)\left(\left(x-3\right)\left(x-1\right)\right)^{2}}{18}$$

Solution Steps

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

$$\frac{1}{18}\left(x+2\right)\left(x^{2}-2x+1\right)\left(x-3\right)^{2}$$

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

$$\frac{1}{18}\left(x+2\right)\left(x^{2}-2x+1\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply \frac{1}{18} by x+2.

$$\left(\frac{1}{18}x+\frac{1}{9}\right)\left(x^{2}-2x+1\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply \frac{1}{18}x+\frac{1}{9} by x^{2}-2x+1 and combine like terms.

$$\left(\frac{1}{18}x^{3}-\frac{1}{6}x+\frac{1}{9}\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply \frac{1}{18}x^{3}-\frac{1}{6}x+\frac{1}{9} by x^{2}-6x+9 and combine like terms.

$$\frac{1}{18}x^{5}-\frac{1}{3}x^{4}+\frac{1}{3}x^{3}+\frac{10}{9}x^{2}-\frac{13}{6}x+1$$

Expand

$$\frac{x^{5}}{18}-\frac{x^{4}}{3}+\frac{x^{3}}{3}+\frac{10x^{2}}{9}-\frac{13x}{6}+1$$

Solution Steps

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

$$\frac{1}{18}\left(x+2\right)\left(x^{2}-2x+1\right)\left(x-3\right)^{2}$$

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

$$\frac{1}{18}\left(x+2\right)\left(x^{2}-2x+1\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply \frac{1}{18} by x+2.

$$\left(\frac{1}{18}x+\frac{1}{9}\right)\left(x^{2}-2x+1\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply \frac{1}{18}x+\frac{1}{9} by x^{2}-2x+1 and combine like terms.

$$\left(\frac{1}{18}x^{3}-\frac{1}{6}x+\frac{1}{9}\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply \frac{1}{18}x^{3}-\frac{1}{6}x+\frac{1}{9} by x^{2}-6x+9 and combine like terms.

$$\frac{1}{18}x^{5}-\frac{1}{3}x^{4}+\frac{1}{3}x^{3}+\frac{10}{9}x^{2}-\frac{13}{6}x+1$$