Adding And Simplifying Fractions With Polynomials

by Andrew McMorgan 50 views

Hey guys! Today, we're diving into the exciting world of adding fractions where the denominators are polynomials. Specifically, we're going to tackle the problem: 5a2+16a+64+9a2−64\frac{5}{a^2+16 a+64}+\frac{9}{a^2-64}. Sounds fun, right? Don't worry; we'll break it down step by step so it's super easy to follow. So, grab your favorite beverage, and let's get started!

Understanding the Problem

Before we jump into adding these fractions, it's essential to understand what we're working with. We have two fractions, each with a polynomial in the denominator. Our goal is to add these two fractions together and, if possible, simplify the result. This involves finding a common denominator, combining the numerators, and then simplifying the resulting fraction.

Factoring the Denominators: The first crucial step in adding these fractions is to factor the denominators. Factoring helps us identify common factors, which is essential for finding the least common denominator (LCD). Let’s factor each denominator separately.

The first denominator is a2+16a+64a^2 + 16a + 64. Notice that this is a quadratic expression. We're looking for two numbers that multiply to 64 and add up to 16. Those numbers are 8 and 8. Therefore, we can factor this expression as (a+8)(a+8)(a+8)(a+8), which is the same as (a+8)2(a+8)^2.

The second denominator is a2−64a^2 - 64. This is a difference of squares, which can be factored as (a−8)(a+8)(a-8)(a+8). Remember, the difference of squares formula is x2−y2=(x−y)(x+y)x^2 - y^2 = (x-y)(x+y).

So now we have:

5(a+8)2+9(a−8)(a+8)\frac{5}{(a+8)^2} + \frac{9}{(a-8)(a+8)}

Finding the Least Common Denominator (LCD)

Now that we've factored the denominators, we need to find the least common denominator (LCD). The LCD is the smallest expression that both denominators can divide into evenly. In this case, we need to consider all unique factors from both denominators.

We have the factors (a+8)2(a+8)^2 and (a−8)(a+8)(a-8)(a+8). The LCD must include each factor to the highest power it appears in either denominator. So, the LCD will be (a+8)2(a−8)(a+8)^2(a-8). This ensures that both (a+8)2(a+8)^2 and (a−8)(a+8)(a-8)(a+8) can divide into the LCD without leaving any remainders. Basically, you want the expression that both denominators can turn into by multiplying by some other expression.

Rewriting the Fractions with the LCD

Next, we need to rewrite each fraction with the LCD as its denominator. This involves multiplying both the numerator and the denominator of each fraction by the appropriate factor to achieve the LCD.

For the first fraction, 5(a+8)2\frac{5}{(a+8)^2}, we need to multiply both the numerator and denominator by (a−8)(a-8) to get the LCD of (a+8)2(a−8)(a+8)^2(a-8):

5(a+8)2⋅(a−8)(a−8)=5(a−8)(a+8)2(a−8)\frac{5}{(a+8)^2} \cdot \frac{(a-8)}{(a-8)} = \frac{5(a-8)}{(a+8)^2(a-8)}

For the second fraction, 9(a−8)(a+8)\frac{9}{(a-8)(a+8)}, we need to multiply both the numerator and denominator by (a+8)(a+8) to get the LCD of (a+8)2(a−8)(a+8)^2(a-8):

9(a−8)(a+8)⋅(a+8)(a+8)=9(a+8)(a+8)2(a−8)\frac{9}{(a-8)(a+8)} \cdot \frac{(a+8)}{(a+8)} = \frac{9(a+8)}{(a+8)^2(a-8)}

Now we have:

5(a−8)(a+8)2(a−8)+9(a+8)(a+8)2(a−8)\frac{5(a-8)}{(a+8)^2(a-8)} + \frac{9(a+8)}{(a+8)^2(a-8)}

Adding the Fractions

Now that both fractions have the same denominator, we can add them together. We do this by adding the numerators and keeping the common denominator:

5(a−8)+9(a+8)(a+8)2(a−8)\frac{5(a-8) + 9(a+8)}{(a+8)^2(a-8)}

Let's simplify the numerator:

5(a−8)+9(a+8)=5a−40+9a+72=14a+325(a-8) + 9(a+8) = 5a - 40 + 9a + 72 = 14a + 32

So our expression becomes:

14a+32(a+8)2(a−8)\frac{14a + 32}{(a+8)^2(a-8)}

Simplifying the Result

Now, let's see if we can simplify the resulting fraction. First, we can factor out a 2 from the numerator:

2(7a+16)(a+8)2(a−8)\frac{2(7a + 16)}{(a+8)^2(a-8)}

Now, we need to check if there are any common factors between the numerator and the denominator. In this case, there are no common factors that we can cancel out. The expression (7a+16)(7a + 16) does not share any factors with (a+8)(a+8) or (a−8)(a-8).

Therefore, the simplified form of our expression is:

2(7a+16)(a+8)2(a−8)\frac{2(7a + 16)}{(a+8)^2(a-8)}

Final Answer

So, after adding the fractions and simplifying, we arrive at the final answer:

2(7a+16)(a+8)2(a−8)\frac{2(7a + 16)}{(a+8)^2(a-8)}

Expanded Form: If you prefer to see the denominator in its expanded form, we can expand (a+8)2(a−8)(a+8)^2(a-8):

(a+8)2(a−8)=(a2+16a+64)(a−8)=a3+16a2+64a−8a2−128a−512=a3+8a2−64a−512(a+8)^2(a-8) = (a^2 + 16a + 64)(a-8) = a^3 + 16a^2 + 64a - 8a^2 - 128a - 512 = a^3 + 8a^2 - 64a - 512

So the fraction can also be written as:

2(7a+16)a3+8a2−64a−512\frac{2(7a + 16)}{a^3 + 8a^2 - 64a - 512}

However, the factored form is generally preferred because it's easier to work with and provides more insight into the expression's structure.

Conclusion

And there you have it! We've successfully added the fractions 5a2+16a+64+9a2−64\frac{5}{a^2+16 a+64}+\frac{9}{a^2-64} and simplified the result. Remember, the key steps are:

  1. Factor the denominators.
  2. Find the least common denominator (LCD).
  3. Rewrite the fractions with the LCD.
  4. Add the numerators.
  5. Simplify the result.

By following these steps, you can confidently tackle similar problems. Keep practicing, and you'll become a pro at adding and simplifying fractions with polynomial denominators. Until next time, keep exploring the fascinating world of mathematics! Cheers, guys!