Example

3(r+2s)=2t-4
a⋅(b-2)=3b
3(r+2s)=2t-4
a⋅(b-2)=3b

Question

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

Answer

$$-3*x^2+3*x^3-x^2+x-3;(3*x^2*(x-1))/((2*x+3)*(x-1));2*x^2-3-1$$

Solution


Factor out the common term \(3{x}^{2}\).
\[\begin{aligned}&-3{x}^{2}+3{x}^{3}-{x}^{2}+x-3\\&\frac{3{x}^{2}(x-1)}{2{x}^{2}+x-3}\\&2{x}^{2}-3-1\end{aligned}\]
Split the second term in \(2{x}^{2}+x-3\) into two terms.
\[\begin{aligned}&-3{x}^{2}+3{x}^{3}-{x}^{2}+x-3\\&\frac{3{x}^{2}(x-1)}{2{x}^{2}+3x-2x-3}\\&2{x}^{2}-3-1\end{aligned}\]
Factor out common terms in the first two terms, then in the last two terms.
\[\begin{aligned}&-3{x}^{2}+3{x}^{3}-{x}^{2}+x-3\\&\frac{3{x}^{2}(x-1)}{x(2x+3)-(2x+3)}\\&2{x}^{2}-3-1\end{aligned}\]
Factor out the common term \(2x+3\).
\[\begin{aligned}&-3{x}^{2}+3{x}^{3}-{x}^{2}+x-3\\&\frac{3{x}^{2}(x-1)}{(2x+3)(x-1)}\\&2{x}^{2}-3-1\end{aligned}\]
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