Question

what is the product?
\\(\frac{2a - 7}{a} cdot \frac{3a^2}{2a^2 - 11a + 14}\\)
options:
\\(\frac{3}{a + 2}\\), \\(\frac{3a}{a + 2}\\), \\(\frac{3}{a - 2}\\), \\(\frac{3a}{a - 2}\\)

Explanation:

Step1: Factor the denominator

Factor \(2a^2 - 11a + 14\). We need two numbers that multiply to \(2\times14 = 28\) and add to \(-11\). The numbers are \(-7\) and \(-4\). So, \(2a^2 - 11a + 14=2a^2 - 7a - 4a + 14=a(2a - 7)-2(2a - 7)=(2a - 7)(a - 2)\).

Step2: Simplify the fraction

The original expression is \(\frac{2a - 7}{a}\cdot\frac{3a^2}{(2a - 7)(a - 2)}\). Cancel out the common factors \((2a - 7)\) and one \(a\) from the numerator and denominator. We get \(\frac{3a}{a - 2}\).

Answer:

\(\boldsymbol{\frac{3a}{a - 2}}\) (the fourth option)