To find the sum of the given rational expressions, we need to add their numerators together over the common denominator. The expressions provided are:
[tex]\[
\frac{2x + 1}{x - 3} + \frac{x}{x - 3} + \frac{4}{x - 3} + \frac{8x^2 + 4x}{x - 3} + \frac{x + 5}{x - 3} + \frac{3x + 5}{x - 3}
\][/tex]
We'll begin by identifying and isolating the numerators of each fraction and then sum them up.
The numerators and their common denominator are:
- [tex]\(2x + 1\)[/tex]
- [tex]\(x\)[/tex]
- [tex]\(4\)[/tex]
- [tex]\(8x^2 + 4x\)[/tex]
- [tex]\(x + 5\)[/tex]
- [tex]\(3x + 5\)[/tex]
Now, combine all these numerators into a single expression:
[tex]\[
(2x + 1) + x + 4 + (8x^2 + 4x) + (x + 5) + (3x + 5)
\][/tex]
Next, let's combine like terms. First, sum all the [tex]\(x^2\)[/tex] terms:
[tex]\[
8x^2
\][/tex]
Then sum all the [tex]\(x\)[/tex] terms:
[tex]\[
2x + x + 4x + x + 3x = 11x
\][/tex]
Next, sum all the constant terms:
[tex]\[
1 + 4 + 5 + 5 = 15
\][/tex]
Now we combine all of these:
[tex]\[
8x^2 + 11x + 15
\][/tex]
The common denominator for all of these expressions is [tex]\(x - 3\)[/tex], so the sum of the given rational expressions is:
[tex]\[
\frac{8x^2 + 11x + 15}{x - 3}
\][/tex]
Therefore, the combined rational expression is:
[tex]\[
\frac{8x^2 + 11x + 15}{x-3}
\][/tex]
