Understanding the Concept of Partial Fractions

Oct 13, 2025 by windailys.com 7 views

Introduction

Partial fractions decomposition is a powerful technique used to simplify complex rational expressions into more manageable components. This method involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate, differentiate, or solve. In this article, we will explore the concept of partial fractions decomposition and apply it to the given expression $\frac{7x^2 - 40x + 52}{x^2 - 6x + 9}$ .

Understanding the Concept of Partial Fractions

Partial fractions decomposition is based on the idea of expressing a rational function as a sum of simpler fractions. The general form of a partial fractions decomposition is:

\frac{P(x)}{Q(x)} = A + \frac{B}{(x - r)^n} + \frac{C}{(x - s)^m} + \cdots

where $P(x)$ and $Q(x)$ are polynomials, $A$ is a constant, $B$ , $C$ , etc. are constants, and $r$ , $s$ , etc. are roots of the denominator polynomial $Q(x)$ .

The Given Expression

The given expression is $\frac{7x^2 - 40x + 52}{x^2 - 6x + 9}$ . To decompose this expression into partial fractions, we need to factor the denominator and express it as a sum of simpler fractions.

Factoring the Denominator

The denominator of the given expression is $x^2 - 6x + 9$ . We can factor this quadratic expression as:

(x - 3)^2

Expressing the Given Expression as Partial Fractions

Now that we have factored the denominator, we can express the given expression as partial fractions. We will assume that the partial fractions decomposition has the form:

\frac{7x^2 - 40x + 52}{(x - 3)^2} = A + \frac{B}{(x - 3)^2}

Finding the Value of A

To find the value of $A$ , we can multiply both sides of the equation by $(x - 3)^2$ :

7x^2 - 40x + 52 = A(x - 3)^2 + B

Finding the Value of B

To find the value of $B$ , we can substitute $x = 3$ into the equation:

7(3)^2 - 40(3) + 52 = A(3 - 3)^2 + B

Simplifying the equation, we get:

7(9) - 40(3) + 52 = A(0)^2 + B

63 - 120 + 52 = B

-5 = B

Finding the Value of A

Now that we have found the value of $B$ , we can substitute it back into the equation:

7x^2 - 40x + 52 = A(x - 3)^2 - 5

We can also substitute $x = 3$ into the equation:

7(3)^2 - 40(3) + 52 = A(3 - 3)^2 - 5

Simplifying the equation, we get:

7(9) - 40(3) + 52 = A(0)^2 - 5

63 - 120 + 52 = -5

-5 = -5

This confirms that our value of $B$ is correct.

Finding the Value of A

Now that we have confirmed the value of $B$ , we can substitute it back into the equation:

7x^2 - 40x + 52 = A(x - 3)^2 - 5

We can also substitute $x = 0$ into the equation:

7(0)^2 - 40(0) + 52 = A(0 - 3)^2 - 5

Simplifying the equation, we get:

52 = A(-3)^2 - 5

52 = 9A - 5

57 = 9A

A = \frac{57}{9}

A = \frac{19}{3}

Conclusion

In this article, we have decomposed the given expression $\frac{7x^2 - 40x + 52}{x^2 - 6x + 9}$ into partial fractions. We have found the values of $A$ and $B$ and expressed the given expression as:

\frac{7x^2 - 40x + 52}{(x - 3)^2} = \frac{19}{3} + \frac{-5}{(x - 3)^2}

This is the correct form for the quotient and remainder, written as partial fractions, of the given expression.

Final Answer

The final answer is:

\boxed{\frac{19}{3} + \frac{-5}{(x - 3)^2}}$$<br/> # **Partial Fractions Decomposition: A Q&A Guide**

Introduction

Partial fractions decomposition is a powerful technique used to simplify complex rational expressions into more manageable components. In our previous article, we explored the concept of partial fractions decomposition and applied it to the given expression $\frac{7x^2 - 40x + 52}{x^2 - 6x + 9}$ . In this article, we will answer some frequently asked questions about partial fractions decomposition.

Q: What is partial fractions decomposition?

A: Partial fractions decomposition is a technique used to simplify complex rational expressions into more manageable components. It involves expressing a rational function as a sum of simpler fractions.

Q: How do I know if a rational expression can be decomposed into partial fractions?

A: A rational expression can be decomposed into partial fractions if and only if the denominator can be factored into linear or quadratic factors.

Q: What are the steps to decompose a rational expression into partial fractions?

A: The steps to decompose a rational expression into partial fractions are:

Factor the denominator.
Express the rational expression as a sum of simpler fractions.
Find the values of the constants in the partial fractions decomposition.

Q: How do I find the values of the constants in the partial fractions decomposition?

A: To find the values of the constants in the partial fractions decomposition, you can use the following methods:

Equate the numerator of the rational expression to the numerator of the partial fractions decomposition.
Solve for the constants using algebraic manipulations.

Q: What are some common mistakes to avoid when decomposing a rational expression into partial fractions?

A: Some common mistakes to avoid when decomposing a rational expression into partial fractions are:

Not factoring the denominator correctly.
Not expressing the rational expression as a sum of simpler fractions correctly.
Not solving for the constants correctly.

Q: Can partial fractions decomposition be used to simplify complex rational expressions?

A: Yes, partial fractions decomposition can be used to simplify complex rational expressions. It involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate, differentiate, or solve.

Q: Are there any limitations to partial fractions decomposition?

A: Yes, there are some limitations to partial fractions decomposition. It can only be used to simplify rational expressions with linear or quadratic denominators. It cannot be used to simplify rational expressions with polynomial denominators of degree 3 or higher.

Q: Can partial fractions decomposition be used to solve differential equations?

A: Yes, partial fractions decomposition can be used to solve differential equations. It involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate or differentiate.

Q: Are there any other applications of partial fractions decomposition?

A: Yes, there are many other applications of partial fractions decomposition. It can be used to simplify complex rational expressions in physics, engineering, and other fields.

Conclusion

In this article, we have answered some frequently asked questions about partial fractions decomposition. We have discussed the concept of partial fractions decomposition, the steps to decompose a rational expression into partial fractions, and some common mistakes to avoid. We have also discussed the limitations and applications of partial fractions decomposition.

Final Answer

The final answer is:

Partial fractions decomposition is a technique used to simplify complex rational expressions into more manageable components.
A rational expression can be decomposed into partial fractions if and only if the denominator can be factored into linear or quadratic factors.
The steps to decompose a rational expression into partial fractions are: factor the denominator, express the rational expression as a sum of simpler fractions, and find the values of the constants in the partial fractions decomposition.
Partial fractions decomposition can be used to simplify complex rational expressions, solve differential equations, and has many other applications.