Question

34 mplify the expression and state the domai 35
\\(\frac{2x^2 + x - 15}{x^2 - x - 12}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(2x^2 + x - 15\): We need two numbers that multiply to \(2\times(-15)= -30\) and add to \(1\). Those numbers are \(6\) and \(-5\). So, \(2x^2 + 6x - 5x - 15 = 2x(x + 3) - 5(x + 3) = (2x - 5)(x + 3)\).

Factor \(x^2 - x - 12\): We need two numbers that multiply to \(-12\) and add to \(-1\). Those numbers are \(-4\) and \(3\). So, \(x^2 - 4x + 3x - 12 = x(x - 4) + 3(x - 4) = (x - 4)(x + 3)\).

Now the expression becomes \(\frac{(2x - 5)(x + 3)}{(x - 4)(x + 3)}\).

Step2: Cancel common factors

Cancel out the common factor \((x + 3)\) (note that \(x
eq -3\) because it would make the original denominator zero). So the simplified expression is \(\frac{2x - 5}{x - 4}\).

Step3: Determine the domain

The original denominator is \(x^2 - x - 12=(x - 4)(x + 3)\). The denominator cannot be zero, so \(x - 4
eq 0\) and \(x + 3
eq 0\), which means \(x
eq 4\) and \(x
eq -3\). So the domain is all real numbers except \(x = -3\) and \(x = 4\).

Answer:

Simplified expression: \(\frac{2x - 5}{x - 4}\) (where \(x
eq -3, 4\)); Domain: All real numbers except \(x = -3\) and \(x = 4\)