To simplify the expression [tex]\(\frac{6 x^3 + 13 x^2 - 5 x}{4 x^2 - 25}\)[/tex], let's break it down step by step.
1. Factor the denominator:
The denominator is [tex]\(4 x^2 - 25\)[/tex], which is a difference of squares. This can be factored as:
[tex]\[
4 x^2 - 25 = (2x)^2 - 5^2 = (2x - 5)(2x + 5)
\][/tex]
2. Factor the numerator:
To simplify the numerator, [tex]\(6 x^3 + 13 x^2 - 5 x\)[/tex], we look for any common factors or possible factorizations. Note that [tex]\(x\)[/tex] is a common factor:
[tex]\[
6 x^3 + 13 x^2 - 5 x = x (6 x^2 + 13 x - 5)
\][/tex]
Now let’s factor the quadratic expression [tex]\(6 x^2 + 13 x - 5\)[/tex]. We look for two numbers that multiply to [tex]\(6 \cdot (-5) = -30\)[/tex] and add up to [tex]\(13\)[/tex]. These numbers are [tex]\(15\)[/tex] and [tex]\(-2\)[/tex]:
[tex]\[
6 x^2 + 13 x - 5 = 6 x^2 + 15 x - 2 x - 5
\][/tex]
Group the terms:
[tex]\[
= (6 x^2 + 15 x) + (-2 x - 5)
\][/tex]
Factor by grouping:
[tex]\[
= 3 x (2 x + 5) - 1 (2 x + 5)
\][/tex]
Factor out the common term:
[tex]\[
= (3 x - 1)(2 x + 5)
\][/tex]
So the numerator [tex]\(6 x^3 + 13 x^2 - 5 x\)[/tex] factors to:
[tex]\[
x (3 x - 1)(2 x + 5)
\][/tex]
3. Simplify the expression:
Now substitute the factored forms of the numerator and the denominator into the original expression:
[tex]\[
\frac{x (3 x - 1)(2 x + 5)}{(2 x - 5)(2 x + 5)}
\][/tex]
Cancel out the common factor of [tex]\((2 x + 5)\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{x (3 x - 1) \cancel{(2 x + 5)}}{(2 x - 5) \cancel{(2 x + 5)}} = \frac{x (3 x - 1)}{2 x - 5}
\][/tex]
4. Final result:
The simplified expression is:
[tex]\[
\frac{x (3 x - 1)}{2 x - 5}
\][/tex]
Thus, [tex]\(\frac{6 x^3 + 13 x^2 - 5 x}{4 x^2 - 25}\)[/tex] simplifies to [tex]\(\frac{x (3 x - 1)}{2 x - 5}\)[/tex].
