Partial Fraction Decomposition: Solve The Expression

Oct 20, 2025 by Dimemap Team 53 views

$Partial Fraction Decomposition: How to Solve $\frac{6x^2 + 12x - 177}{x^2 + x - 30}$$

Hey guys! Today, we're diving into the fascinating world of partial fraction decomposition, specifically focusing on how to tackle improper rational expressions. We'll break down the steps to solve the expression $\frac{6x^2 + 12x - 177}{x^2 + x - 30}$ . Trust me, once you get the hang of it, it's like solving a puzzle! So, let's jump right in!

Understanding Partial Fraction Decomposition

Before we dive into the specifics of our problem, let’s make sure we’re all on the same page about what partial fraction decomposition actually is. In simple terms, it's a technique used to break down a complex rational expression (a fraction where the numerator and denominator are polynomials) into simpler fractions. This is super helpful in calculus, differential equations, and even some areas of engineering. Imagine trying to integrate a complicated fraction – it can be a nightmare! But if you decompose it into simpler fractions, the integration becomes much more manageable.

So, why does this work? The basic idea is that any rational expression can be expressed as a sum of fractions with simpler denominators. Think of it like breaking down a recipe into its individual ingredients. Each simpler fraction is a “partial fraction,” and putting them together gives you the original expression. The technique is particularly useful when dealing with improper rational expressions, which is what we're focusing on today. An improper rational expression is one where the degree of the numerator is greater than or equal to the degree of the denominator. This is crucial because we need to perform long division first to simplify the expression before we can decompose it into partial fractions.

Let's highlight some key terms to keep in mind:

Rational Expression: A fraction where both the numerator and the denominator are polynomials.
Partial Fraction: A simpler fraction that, when added together with other partial fractions, forms the original rational expression.
Improper Rational Expression: A rational expression where the degree of the numerator is greater than or equal to the degree of the denominator.
Decomposition: The process of breaking down a complex fraction into simpler ones.

Understanding these concepts is the first step in mastering partial fraction decomposition. Now that we've got the basics down, let's move on to the problem at hand and see how these concepts come into play!

Step 1: Polynomial Long Division

Alright, let's tackle our expression: $\frac{6x^2 + 12x - 177}{x^2 + x - 30}$ . As we've established, this is an improper rational expression because the degree of the numerator (2) is equal to the degree of the denominator (2). This means our first step is to perform polynomial long division. I know, I know, it might sound intimidating, but trust me, it's not as scary as it seems. We're essentially dividing the numerator by the denominator to simplify things.

Think back to your basic long division skills with numbers. Polynomial long division is quite similar, just with variables and exponents involved. We're trying to find out how many times the denominator ( $x^2 + x - 30$ ) goes into the numerator ( $6x^2 + 12x - 177$ ). So, how do we do it?

Set up the long division: Write the denominator ( $x^2 + x - 30$ ) outside the division symbol and the numerator ( $6x^2 + 12x - 177$ ) inside.
Divide the leading terms: Divide the leading term of the numerator ( $6x^2$ ) by the leading term of the denominator ( $x^2$ ). This gives us 6. This is the first term of our quotient (the result of the division).
Multiply: Multiply the entire denominator ( $x^2 + x - 30$ ) by the quotient term (6). This gives us $6x^2 + 6x - 180$ .
Subtract: Subtract the result ( $6x^2 + 6x - 180$ ) from the numerator ( $6x^2 + 12x - 177$ ). This gives us $(6x^2 + 12x - 177) - (6x^2 + 6x - 180) = 6x + 3$ .
Remainder: The result of the subtraction, $6x + 3$ , is our remainder. Since the degree of the remainder (1) is less than the degree of the denominator (2), we stop the long division process.

So, after performing the long division, we find that:

$\frac{6x^2 + 12x - 177}{x^2 + x - 30} = 6 + \frac{6x + 3}{x^2 + x - 30}$

This is a crucial step because we've now separated the original improper fraction into a whole number (6) and a proper fraction ( $\frac{6x + 3}{x^2 + x - 30}$ ). We'll be working with this proper fraction in the next steps. This simplification makes the partial fraction decomposition process much easier. Remember, the goal here is to break down the complex expression into simpler parts that are easier to handle. Polynomial long division is the key to achieving this when dealing with improper rational expressions.

Step 2: Factor the Denominator

Now that we've handled the long division and have our simplified expression, $6 + \frac{6x + 3}{x^2 + x - 30}$ , it’s time to focus on the fractional part. The next crucial step in partial fraction decomposition is to factor the denominator. Factoring the denominator allows us to identify the simpler factors that will become the denominators of our partial fractions. This step is essential because it lays the foundation for breaking down the complex fraction into its simpler components.

In our case, the denominator is $x^2 + x - 30$ . We need to find two binomials that multiply together to give us this quadratic expression. Think of it like reversing the FOIL method (First, Outer, Inner, Last) that you might have learned in algebra. We're looking for two numbers that multiply to -30 and add up to 1 (the coefficient of the x term). So, what are those numbers?

After a bit of thought (or maybe a quick mental calculation), you'll realize that the numbers are 6 and -5. This is because 6 * -5 = -30 and 6 + (-5) = 1. Therefore, we can factor the denominator as follows:

$x^2 + x - 30 = (x + 6)(x - 5)$

Great! We've successfully factored the denominator. This is a significant step because it tells us the form our partial fractions will take. Since we have two distinct linear factors (x + 6) and (x - 5), we'll have two partial fractions, each with one of these factors as its denominator.

To recap, factoring the denominator is a critical step in partial fraction decomposition. It allows us to identify the building blocks of our partial fractions and sets the stage for the next steps in the process. Remember, the more comfortable you become with factoring, the easier this whole process will be. Now that we have our factored denominator, let's move on to setting up the partial fraction decomposition.

Step 3: Set Up the Partial Fraction Decomposition

Okay, with our denominator factored as $(x + 6)(x - 5)$ , we're ready to set up the partial fraction decomposition. This is where we express the proper fraction $\frac{6x + 3}{(x + 6)(x - 5)}$ as a sum of simpler fractions, each with one of the factors from the denominator as its denominator. This step is like creating the blueprint for our final solution – we're defining the structure of the decomposed fractions.

Since we have two distinct linear factors, $(x + 6)$ and $(x - 5)$ , we'll have two partial fractions. Each fraction will have a constant numerator (which we'll call A and B) and one of the factors as its denominator. So, we can write the decomposition as follows:

$\frac{6x + 3}{(x + 6)(x - 5)} = \frac{A}{x + 6} + \frac{B}{x - 5}$

Here, A and B are constants that we need to determine. Our goal is to find the values of A and B that make this equation true. Once we find these values, we'll have successfully decomposed the original fraction into its partial fractions.

The equation above is the heart of the partial fraction decomposition. It states that our original complex fraction can be expressed as the sum of two simpler fractions. The constants A and B are the unknowns we need to solve for. There are a couple of common methods for finding these constants, which we'll explore in the next step.

Setting up the partial fraction decomposition correctly is crucial for the rest of the process. It ensures that we have the right framework for finding the partial fractions. Remember, each factor in the denominator will correspond to a partial fraction, and the numerators will be constants (for linear factors). Now that we have our setup, let's move on to the exciting part: solving for A and B!

Step 4: Solve for the Unknown Constants

Now for the fun part: solving for the unknown constants, A and B, in our partial fraction decomposition. We've established that $\frac{6x + 3}{(x + 6)(x - 5)} = \frac{A}{x + 6} + \frac{B}{x - 5}$ . There are a couple of common methods to find the values of A and B: the method of equating coefficients and the method of substituting values. Let's explore the method of substituting values, as it's often quicker and more straightforward for this type of problem.

The method of substituting values involves choosing strategic values for x that will eliminate one of the unknowns, allowing us to solve for the other. Since our denominators are (x + 6) and (x - 5), the values x = -6 and x = 5 are excellent choices because they will make one of the denominators zero.

Substitute x = -6:

Plugging x = -6 into our equation, we get:

$\frac{6(-6) + 3}{(-6 + 6)(-6 - 5)} = \frac{A}{-6 + 6} + \frac{B}{-6 - 5}$

$\frac{-36 + 3}{0*(-11)} = \frac{A}{0} + \frac{B}{-11}$

$\frac{-33}{(-6 + 6)(-6 - 5)} = \frac{A}{(-6 + 6)} + \frac{B}{(-6 - 5)}$

$\frac{-33}{0} = \frac{A}{0} + \frac{B}{-11}$

Wait a minute! Multiplying both sides by $(-6+6)(-6-5)$ to clear the denominators, we get:

$6(-6) + 3 = A(-6 - 5) + B(-6 + 6)$

$-33 = A(-11) + B(0)$

$-33 = -11A$

Dividing both sides by -11, we find A = 3.

Substitute x = 5:

Now, let's plug in x = 5:

$\frac{6(5) + 3}{(5 + 6)(5 - 5)} = \frac{A}{5 + 6} + \frac{B}{5 - 5}$

$\frac{30 + 3}{(11)(0)} = \frac{A}{11} + \frac{B}{0}$

Again, to avoid division by zero, multiply both sides by $(5+6)(5-5)$ :

$6(5) + 3 = A(5 - 5) + B(5 + 6)$

$33 = A(0) + B(11)$

$33 = 11B$

Dividing both sides by 11, we get B = 3.

So, we've successfully found that A = 3 and B = 3. This is a major breakthrough! We now know the values of the constants in our partial fraction decomposition.

Solving for the unknown constants is the heart of the partial fraction decomposition process. The method of substituting values is a powerful tool that allows us to isolate and solve for each constant efficiently. With A and B determined, we're just one step away from the final answer. Let's put it all together in the final step!

Step 5: Write the Final Decomposition

Alright, we've reached the final stretch! We've done the polynomial long division, factored the denominator, set up the partial fraction decomposition, and solved for the unknown constants. Now, it's time to write the final decomposition. This is where we put all the pieces together to express the original improper rational expression as a sum of simpler fractions.

Remember, we started with the expression $\frac{6x^2 + 12x - 177}{x^2 + x - 30}$ . After polynomial long division, we got $6 + \frac{6x + 3}{x^2 + x - 30}$ . Then, we factored the denominator and set up the partial fraction decomposition as $\frac{6x + 3}{(x + 6)(x - 5)} = \frac{A}{x + 6} + \frac{B}{x - 5}$ . Finally, we solved for A and B and found that A = 3 and B = 3.

Now, we can substitute the values of A and B back into our partial fraction decomposition:

$\frac{6x + 3}{(x + 6)(x - 5)} = \frac{3}{x + 6} + \frac{3}{x - 5}$

Don't forget about the whole number part we got from the long division! We need to include that in our final answer. So, the complete partial fraction decomposition is:

$\frac{6x^2 + 12x - 177}{x^2 + x - 30} = 6 + \frac{3}{x + 6} + \frac{3}{x - 5}$

And there you have it! We've successfully decomposed the improper rational expression into a sum of simpler fractions. This is the final answer, and it represents the original expression broken down into its partial fraction components.

Writing the final decomposition is the culmination of all our hard work. It's the moment where we see the original complex expression transformed into a more manageable form. Remember to include any whole number parts from the long division and to substitute the values of the constants you solved for. Congratulations, you've mastered partial fraction decomposition!

Conclusion

So there you have it, guys! We've walked through the entire process of finding the partial fraction decomposition of the improper rational expression $\frac{6x^2 + 12x - 177}{x^2 + x - 30}$ . We covered everything from polynomial long division to factoring the denominator, setting up the decomposition, solving for the constants, and finally, writing out the decomposed form. This is a powerful technique that's super useful in various areas of math and engineering.

Remember, the key to mastering partial fraction decomposition is practice. The more you work through problems like this, the more comfortable you'll become with each step. Don't be afraid to tackle challenging expressions and to review the steps as needed. With a little bit of effort, you'll be decomposing rational expressions like a pro!

If you found this guide helpful, be sure to share it with your friends and classmates. And if you have any questions or want to dive deeper into this topic, feel free to leave a comment below. Keep practicing, keep exploring, and keep conquering those math challenges! You've got this!