Simplify 5/(5c^2+9c+4) + 7/(c+1): A Step-by-Step Guide

Hey guys! Today, we are going to dive into simplifying a rational expression. This involves factoring, finding common denominators, and combining like terms. It might sound like a mouthful, but don't worry, we'll break it down step by step so it's super easy to follow. Our main goal is to simplify the expression 5/(5c^2+9c+4) + 7/(c+1) as much as possible. This means we're going to factor any factorable parts, find a common denominator, combine the fractions, and then simplify the result. Think of it like putting together a puzzle – each step is a piece that fits into the bigger picture. Let's get started!

Step 1: Factoring the Denominator

The first thing we need to do is factor the quadratic expression in the denominator of the first fraction, which is 5c^2 + 9c + 4. Factoring this quadratic is a crucial step in simplifying the entire expression. To factor it, we need to find two binomials that, when multiplied together, give us the original quadratic. We're looking for two numbers that multiply to (5 * 4 = 20) and add up to 9. Those numbers are 4 and 5. Now, we can rewrite the middle term (9c) using these numbers:

5c^2 + 9c + 4 = 5c^2 + 5c + 4c + 4

Next, we factor by grouping. We group the first two terms and the last two terms together:

(5c^2 + 5c) + (4c + 4)

From the first group, we can factor out a 5c, and from the second group, we can factor out a 4:

5c(c + 1) + 4(c + 1)

Notice that both terms now have a common factor of (c + 1). We can factor this out:

(5c + 4)(c + 1)

So, the factored form of 5c^2 + 9c + 4 is (5c + 4)(c + 1). This is a significant step because it allows us to see common factors between the denominators of our two fractions. Factoring is like finding the basic building blocks of an expression, which makes it much easier to work with. Now that we have the factored form, we can move on to the next step, which involves finding a common denominator.

Step 2: Finding a Common Denominator

Now that we've factored the first denominator, we can rewrite our original expression as:

5 / [(5c + 4)(c + 1)] + 7 / (c + 1)

To add these two fractions, we need to have a common denominator. Looking at the two denominators, (5c + 4)(c + 1) and (c + 1), we can see that the least common denominator (LCD) is (5c + 4)(c + 1). The first fraction already has this denominator, which is great! But the second fraction, 7 / (c + 1), needs to be adjusted. To get the LCD, we need to multiply both the numerator and the denominator of the second fraction by (5c + 4). This ensures that we're not changing the value of the fraction, just its form. So, we multiply like this:

[7 / (c + 1)] * [(5c + 4) / (5c + 4)] = [7(5c + 4)] / [(5c + 4)(c + 1)]

This gives us:

(35c + 28) / [(5c + 4)(c + 1)]

Now, both fractions have the same denominator, which means we can add them together. Finding a common denominator is like making sure everyone speaks the same language – it allows us to combine things that were previously incompatible. With the common denominator in place, we're ready to move on to adding the fractions and simplifying the result. This is where we'll combine the numerators and see if there are any terms we can simplify further.

Step 3: Adding the Fractions

With the common denominator, we can now add the fractions. Our expression looks like this:

5 / [(5c + 4)(c + 1)] + (35c + 28) / [(5c + 4)(c + 1)]

Since the denominators are the same, we can add the numerators directly:

[5 + (35c + 28)] / [(5c + 4)(c + 1)]

Now, we simplify the numerator by combining like terms:

(35c + 33) / [(5c + 4)(c + 1)]

At this point, we have a single fraction. The next step is to see if we can simplify it further. This usually involves checking if the numerator and denominator have any common factors that can be canceled out. Adding the fractions is like combining ingredients in a recipe – once they're mixed, we can see if the result can be further refined. Now, we move on to the final step: simplifying the fraction.

Step 4: Simplifying the Fraction

We have the expression:

(35c + 33) / [(5c + 4)(c + 1)]

To simplify, we need to see if the numerator and denominator have any common factors. Let's take a look at the numerator, 35c + 33. We can try to factor out the greatest common factor (GCF). The GCF of 35 and 33 is 1, so there isn't a common numerical factor we can extract. We also need to check if 35c + 33 can be factored into a form that includes either (5c + 4) or (c + 1), as those are the factors in the denominator. Unfortunately, 35c + 33 doesn't factor in a way that would allow us to cancel anything with the denominator. This means that the expression is already in its simplest form.

So, the simplified expression is:

(35c + 33) / [(5c + 4)(c + 1)]

Simplifying fractions is like tidying up – we want to make sure everything is in its neatest and most organized form. In this case, we couldn't simplify any further, so we've reached our final answer. Congrats, guys! We've successfully simplified the given expression.

Final Answer

The completely factored and simplified form of the expression 5/(5c^2+9c+4) + 7/(c+1) is:

(35c + 33) / [(5c + 4)(c + 1)]

We started by factoring the denominator, then found a common denominator, added the fractions, and finally simplified the result. Each step was crucial in getting to our final answer. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time! If you have any questions, feel free to ask. Keep up the great work, everyone!