Question

In Exercises $1-4,$ find the values of $A$ and $B$ that complete the partial fraction decomposition. $$\frac{x-12}{x^{2}-4 x}=\frac{A}{x}+\frac{B}{x-4}$$

Name: Problem 1, Chapter 6: Calculus: Graphical, Numerical, Algebraic
Uploaded: 2020-03-24T21:54:27-08:00
Duration: 2 min 45 s
Channel: Bryan Lynn
Description: In Exercises 1-4, find the values of A and B that complete the partial fraction decomposition. (x-12)/(x^2-4 x)=(A)/(x)+(B)/(x-4)

   In Exercises $1-4,$ find the values of $A$ and $B$ that complete the partial fraction decomposition. $$\frac{x-12}{x^{2}-4 x}=\frac{A}{x}+\frac{B}{x-4}$$

Calculus: Graphical, Numerical, Algebraic

Ross L. Finney,… 3rd Edition

Chapter 6, Problem 1

Instant Answer

Solved by Expert Bryan Lynn

03/24/2020

Step 1

In order to add these two fractions together, they would have to share a common denominator. We can do that by multiplying $\frac{A}{x}$ by $\frac{x-4}{x-4}$ and $\frac{B}{x-4}$ by $\frac{x}{x}$, which gives us $\frac{A(x-4)}{x^{2}-4x} + \frac{Bx}{x^{2}-4x}$. Show more…

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In Exercises $1-4,$ find the values of $A$ and $B$ that complete the partial fraction decomposition. $$\frac{x-12}{x^{2}-4 x}=\frac{A}{x}+\frac{B}{x-4}$$

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Key Concepts

Partial Fraction Decomposition

This is a technique for expressing a complex rational expression as a sum of simpler fractions, which can simplify algebraic manipulation and integration. It involves breaking down the original fraction into a sum of fractions with simpler denominators.

Factorization

Factorization involves expressing a polynomial as a product of its factors. In the context of partial fractions, factorizing the denominator is critical as it determines the form of the partial fractions into which the original rational expression is decomposed.

Equating Coefficients

Once the partial fractions are set up with a common denominator, equating coefficients is used to match the terms from the numerator of the decomposed expression to those of the original expression. This process yields a system of equations in the unknown constants, which can be solved to complete the decomposition.

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