 actever6a

2021-11-16

To calculate: The pardital decomposition of the function $\frac{-12x-29}{2{x}^{2}+11x+15}$ Fommeirj

Given Information:
The provided expression is:$\frac{-12x-29}{2{x}^{2}+11x+15}$.
Formula used:
Decomposition of $\frac{f\left(x\right)}{g\left(x\right)}$ into Partial Fractions:
Consider a rational expression $\frac{f\left(x\right)}{g\left(x\right)}$, where f(x) and g(x) are polynomial with real coefficients, $g\left(x\right)\ne 0$, and the degree of f(x) is less than degree of g(x).
Step 1: Factor the denominator g (x) completely into linear factors of the form ${\left(ax+b\right)}^{m}$ and quadratic factors of the form ($a{x}^{2}+bx+c{\right)}^{n}$ that are not further factorable over the integers.
Step 2: Set up the form of decomposition. That is, write the original rational expression $\frac{f\left(x\right)}{g\left(x\right)}$ as a sum of simpler fractions using these guidelines. Note that ${A}_{1},{A}_{2},\dots ..,{A}_{m},{B}_{1},{B}_{2},\dots ..,{B}_{m}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{C}_{1},{C}_{2},\dots .,{C}_{m}$ are constants.
Linear Factors of g(x):
For each linear factor of g(x), the partial fraction decomposition must include the sum:
$\frac{{A}_{1}}{{\left(ax+b\right)}^{1}}+\frac{{A}_{2}}{{\left(ax+b\right)}^{2}}+\dots +\frac{{A}_{m}}{{\left(ax+b\right)}^{m}}$
For each quadratic factor of g (x), the partial fraction decomposition must include the sum:
$\frac{{B}_{1}x+{C}_{1}}{{\left(a{x}^{2}+bx+c\right)}^{1}}+\frac{{B}_{1}x+{C}_{1}}{{\left(a{x}^{2}+bx+c\right)}^{1}}+\dots +\frac{{B}_{1}x+{C}_{1}}{{\left(a{x}^{2}+bx+c\right)}^{1}}$
Step 3: With the form of the partial fraction decomposition set up, multiply both sides of the equation by the Least Common Divisor to clear fractions.
Step 4: Use the equation from step3, set up a system of linear equations by equating the constant terms and equating the coefficients of like powers ofx.
Step 5: Solve the system of equations from step4 and substitute the solutions to the system into the partial fraction decomposition.
Calculation:
Consider the provided expression,
$\frac{-12x-29}{2{x}^{2}+11x+15}$
Here, $f\left(x\right)=-12x—29\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g\left(x\right)=2{x}^{2}+11x+15$
Factorize g(x):
$g\left(x\right)=2{x}^{2}+11x+15$
$=2{x}^{2}+6x+5x+15$
$=2x\left(x+3\right)+5\left(x+3\right)$
$=\left(2x+5\right)\left(x+3\right)$
So, the expression can be written as:

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