91Ó°ÊÓ

A rational function is a fraction where both the numerator and the denominator are polynomials. An example is the function in our exercise: \(\frac{9x^2 + 21x - 24}{x^3 + 4x^2 - 11x - 30}\). These functions are essential in mathematics because they often appear in real-world applications, such as physics and engineering.
To understand a rational function better, we can decompose it into simpler fractions, which makes it easier to analyze and integrate. This process is called partial fraction decomposition. By breaking down a complex rational function into simpler parts, we can solve integrals, differential equations, and other mathematical problems more easily.

To find constants in partial fraction decomposition, we often need to solve a system of equations. Systems of equations are sets of equations with multiple variables that we solve simultaneously. In our example:
\(\begin{align*} A + B + C &= 9,\ 2A + 7B - C &= 21,\ -15A + 10B - 6C &= -24 \end{align*}\)
We found three equations derived from comparing coefficients of like terms.
Solving systems can be done with methods like substitution, elimination, or using matrices. In this case, solving reveals that \(A = 1\), \(B = 2\), and \(C = 6\). Once we have these values, we substitute them back into the partial fractions to reconstruct the original rational function.

Factorization simplifies polynomials into products of simpler polynomials, which is crucial for partial fraction decomposition. For example, the denominator polynomial \((x^3 + 4x^2 - 11x - 30)\) can be factored as \( (x+2)(x-3)(x+5)\). These factors break the polynomial into linear components.
Factoring polynomials involves finding the roots or zeros of the polynomial, which are values of x where the polynomial equals zero. Techniques include synthetic division, long division, and using the quadratic formula for quadratic polynomials.
By factoring the denominator, we express complicated fractions in simpler forms, paving the way to solving for constants in partial fraction decomposition.