Question

simplify the rational expression.
\\(\frac{2x^{2}-7x + 5}{3x^{2}+x - 4}\\)
\\(\frac{2x^{2}-7x + 5}{3x^{2}+x - 4}=\square\\)

Explanation:

Step1: Factor the numerator

For the quadratic expression \(2x^{2}-7x + 5\), we need to find two numbers that multiply to \(2\times5 = 10\) and add up to \(-7\). The numbers are \(-2\) and \(-5\).
So we rewrite the middle term:
\(2x^{2}-2x-5x + 5\)
Group the terms:
\((2x^{2}-2x)-(5x - 5)\)
Factor out the common factors from each group:
\(2x(x - 1)-5(x - 1)\)
Then factor out \((x - 1)\):
\((2x - 5)(x - 1)\)

Step2: Factor the denominator

For the quadratic expression \(3x^{2}+x - 4\), we need to find two numbers that multiply to \(3\times(-4)=-12\) and add up to \(1\). The numbers are \(4\) and \(-3\).
Rewrite the middle term:
\(3x^{2}+4x-3x - 4\)
Group the terms:
\((3x^{2}+4x)-(3x + 4)\)
Factor out the common factors from each group:
\(x(3x + 4)-1(3x + 4)\)
Then factor out \((3x + 4)\):
\((x - 1)(3x + 4)\)

Step3: Simplify the rational expression

Now we have the numerator factored as \((2x - 5)(x - 1)\) and the denominator factored as \((x - 1)(3x + 4)\). We can cancel out the common factor \((x - 1)\) (assuming \(x
eq1\) to avoid division by zero):
\(\frac{(2x - 5)(x - 1)}{(x - 1)(3x + 4)}=\frac{2x - 5}{3x + 4}\)

Answer:

\(\frac{2x - 5}{3x + 4}\)