Question

1. \\(\frac{3x^2 - 13x + 14}{x^2 - 5x + 6}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(3x^2 - 13x + 14\): We need two numbers that multiply to \(3\times14 = 42\) and add to \(-13\). The numbers are \(-6\) and \(-7\). So, \(3x^2 - 6x - 7x + 14 = 3x(x - 2) - 7(x - 2)=(3x - 7)(x - 2)\).

Factor \(x^2 - 5x + 6\): We need two numbers that multiply to \(6\) and add to \(-5\). The numbers are \(-2\) and \(-3\). So, \(x^2 - 2x - 3x + 6 = x(x - 2) - 3(x - 2)=(x - 2)(x - 3)\).

Now the expression becomes \(\frac{(3x - 7)(x - 2)}{(x - 2)(x - 3)}\).

Step2: Cancel common factors

Cancel out the common factor \((x - 2)\) (assuming \(x
eq2\) to avoid division by zero). So we get \(\frac{3x - 7}{x - 3}\).

Answer:

\(\frac{3x - 7}{x - 3}\) (for \(x
eq2\) and \(x
eq3\))