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

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

Solve for y

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

Solution Steps

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

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

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

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

Express 3\times \frac{x^{2}+2x+1}{3} as a single fraction.

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

Cancel out 3 and 3.

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

Use the distributive property to multiply x^{2}+2x+1 by x.

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

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

$$y=x^{6}-x^{5}-2x^{4}+2x^{3}+x^{2}-x$$