Find the partial fraction decomposition for each rational expression. (2)/(x^2-9)

Name: Problem 15, Chapter 8: Precalculus Functions and Graphs
Uploaded: 2020-04-16T23:14:05-08:00
Duration: 1 min 36 s
Channel: Xiaomeng Zhang
Description: Find the partial fraction decomposition for each rational expression. (2)/(x^2-9)

step 1 Here we have x square minus 9 equals to x plus 3 times x minus 3. So we have 2 over x square min

Find the partial fraction decomposition for each rational...

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is the process of breaking down a rational expression into a sum of simpler fractions. This technique is particularly useful when dealing with integration or other operations in calculus, as it transforms complex fractions into a sum of terms that are easier to work with. It typically requires expressing the original fraction as a sum of fractions whose denominators are factors of the original denominator.

Difference of Squares Factorization

The difference of squares is a specific pattern in algebra where an expression can be rewritten as the product of two binomials. In the context of the given problem, the quadratic denominator x² - 9 is recognized as a difference of squares, which factors into (x - 3)(x + 3). This factorization is a crucial step in setting up the partial fractions properly.

Solving for Constants in Decomposition

Once the denominator is factored and the partial fractions have been set up, the next key concept is to solve for the unknown constants in the numerators of the simpler fractions. This is typically done by combining the fractions over a common denominator and equating the numerator to that of the original fraction. Choosing specific values for the variable or comparing coefficients are common methods used to determine these constants.

Transcript

Please give Ace some feedback