Question

question 7 of 10
select the correct answer.
what is the simplest form of this expression?
\\(\frac{15x - 3}{5x^2 - 11x + 2} + \frac{x - 5}{2x^2 - 11x + 5}\\)
\\(\circ\\) \\(\frac{5x - 5}{2x^2 - 3x + 2}\\)
\\(\circ\\) \\(\frac{7x - 5}{2x^2 - 5x + 2}\\)
\\(\circ\\) \\(\frac{4}{2x^2 - 5x - 1}\\)
\\(\circ\\) \\(\frac{3x - 3}{2x^2 - 5x + 2}\\)

Explanation:

Step1: Factor Denominators

Factor \(5x^2 - 11x + 2\): Find two numbers that multiply to \(5\times2 = 10\) and add to \(-11\), which are \(-10\) and \(-1\). So, \(5x^2 - 10x - x + 2 = 5x(x - 2) - 1(x - 2)=(5x - 1)(x - 2)\).

Factor \(2x^2 - 11x + 5\): Find two numbers that multiply to \(2\times5 = 10\) and add to \(-11\), which are \(-10\) and \(-1\). So, \(2x^2 - 10x - x + 5 = 2x(x - 5) - 1(x - 5)=(2x - 1)(x - 5)\).

The expression becomes \(\frac{15x - 3}{(5x - 1)(x - 2)}+\frac{x - 5}{(2x - 1)(x - 5)}\).

Step2: Simplify Fractions

Simplify \(\frac{x - 5}{(2x - 1)(x - 5)}\) (note \(x
eq5\)): Cancel \(x - 5\) to get \(\frac{1}{2x - 1}\).

Factor \(15x - 3\): \(3(5x - 1)\), so \(\frac{15x - 3}{(5x - 1)(x - 2)}=\frac{3(5x - 1)}{(5x - 1)(x - 2)}\) (note \(x
eq\frac{1}{5}\)). Cancel \(5x - 1\) to get \(\frac{3}{x - 2}\).

Now the expression is \(\frac{3}{x - 2}+\frac{1}{2x - 1}\).

Step3: Find Common Denominator

Common denominator is \((x - 2)(2x - 1)=2x^2 - 5x + 2\).

Rewrite fractions: \(\frac{3(2x - 1)}{(x - 2)(2x - 1)}+\frac{1(x - 2)}{(x - 2)(2x - 1)}\).

Step4: Add Numerators

Expand numerators: \(3(2x - 1)=6x - 3\) and \(1(x - 2)=x - 2\).

Add: \(6x - 3 + x - 2 = 7x - 5\).

So the simplified form is \(\frac{7x - 5}{2x^2 - 5x + 2}\).

Answer:

B. \(\frac{7x - 5}{2x^2 - 5x + 2}\)