Question

simplify the expression: \\(\frac{\left(\frac{2x^2 - 7x + 3}{10x^3}\
ight)}{\left(\frac{3 - x}{5}\
ight)}\\) (1 point) \\(\circ\\) \\(\frac{2x^3}{2x - 1}\\) \\(\circ\\) \\(\frac{2x^3}{1 - 2x}\\) \\(\circ\\) \\(\frac{1 - 2x}{2x^3}\\) \\(\circ\\) \\(\frac{2x - 1}{2x^3}\\)

Explanation:

Step1: Factor the quadratic

Factor \(2x^2 - 7x + 3\). We need two numbers that multiply to \(2\times3 = 6\) and add to \(-7\). The numbers are \(-6\) and \(-1\). So, \(2x^2 - 6x - x + 3 = 2x(x - 3) - 1(x - 3)=(2x - 1)(x - 3)\)

Step2: Rewrite the denominator's numerator

Notice that \(3 - x=-(x - 3)\)

Step3: Rewrite the complex fraction as multiplication

Dividing by a fraction is multiplying by its reciprocal. So, \(\frac{\frac{(2x - 1)(x - 3)}{10x^3}}{\frac{3 - x}{5}}=\frac{(2x - 1)(x - 3)}{10x^3}\times\frac{5}{3 - x}\)

Step4: Substitute \(3 - x = -(x - 3)\)

\(\frac{(2x - 1)(x - 3)}{10x^3}\times\frac{5}{-(x - 3)}\)

Step5: Cancel common factors

Cancel \((x - 3)\) (assuming \(x
eq3\)) and simplify \(\frac{5}{10}=\frac{1}{2}\). We get \(\frac{(2x - 1)\times(-1)}{2x^3}=\frac{1 - 2x}{2x^3}\) (or \(\frac{-(2x - 1)}{2x^3}=\frac{1 - 2x}{2x^3}\))

Answer:

\(\boldsymbol{\frac{1 - 2x}{2x^3}}\) (which corresponds to the option \(\frac{1 - 2x}{2x^3}\) among the given choices)