Final answer:
The partial fraction decomposition of the provided expression requires breaking it down into a form involving separate fractions with constants to be determined. The correct decomposition isn't listed among the options but involves a linear term for the quadratic denominator. The process includes multiplying by the common denominator and comparing coefficients or plugging in values to solve for the constants.
Step-by-step explanation:
The partial fraction decomposition of 7x² - 6x + 9 / 3x(4x² + 9) involves breaking down the complex rational expression into simpler fractions that are easier to work with and integrate. The decomposition would generally take the form:
To find A, B, and C, you would multiply both sides of the equation by the denominator so that on one side you have just the numerator and on the other side you have a series of terms that include A, B, and C. This equation can then be solved by comparing coefficients or by substituting values of x that simplify the equation.
As a note, negative exponents denote a division rather than multiplication and are equivalent to putting the base of the exponent into the denominator. This concept is useful in various mathematical operations, such as simplifying expressions and solving equations.
When a quadratic is involved in the denominator as in 4x² + 9, the corresponding part of the partial fraction will likely be of the form (Bx+C) since this quadratic does not factor over the real numbers and you need a linear term to account for the x in the numerator.
Among the given options for partial fractions, none exactly represent the correct initial decomposition. However, if solving for A, B, and C is performed correctly, one should end up with an equivalent expression to the original rational function through the process of partial fraction decomposition.