Solve {r}{2x^2-x+3}{x^2+2x-5}{3x^2-x-3}

$$\left. \begin{array} { r } { 2 x ^ { 2 } - x + 3 } \\ { x ^ { 2 } + 2 x - 5 } \\ { 3 x ^ { 2 } - x - 3 } \end{array} \right.$$

Least Common Multiple

$$\frac{\left(2x^{2}-x+3\right)\left(\left(6x-1\right)^{2}-37\right)\left(\left(x+1\right)^{2}-6\right)}{12}$$

Solution Steps

Factor the expressions that are not already factored.

$$x^{2}+2x-5=\left(x-\left(\sqrt{6}-1\right)\right)\left(x-\left(-\sqrt{6}-1\right)\right) 3x^{2}-x-3=3\left(x-\left(-\frac{1}{6}\sqrt{37}+\frac{1}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{37}+\frac{1}{6}\right)\right)$$

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

$$3\left(x-\left(-\sqrt{6}-1\right)\right)\left(x-\left(\sqrt{6}-1\right)\right)\left(x-\frac{1-\sqrt{37}}{6}\right)\left(x-\frac{\sqrt{37}+1}{6}\right)\left(2x^{2}-x+3\right)$$

Expand the expression.

$$6x^{6}+7x^{5}-36x^{4}+33x^{3}-29x^{2}-18x+45$$

Evaluate

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