To find the sum of the polynomials [tex]\((3x^3 - 5x - 8) + (5x^3 + 7x + 3)\)[/tex], follow these steps:
1. Write down the given polynomials:
- The first polynomial is [tex]\(3x^3 - 5x - 8\)[/tex].
- The second polynomial is [tex]\(5x^3 + 7x + 3\)[/tex].
2. Align the polynomials by their like terms, which are the same powers of [tex]\(x\)[/tex]:
[tex]\[
\begin{array}{r}
3x^3 + 0x^2 - 5x - 8 \\
+ \ 5x^3 + 0x^2 + 7x + 3 \\
\end{array}
\][/tex]
3. Add the like terms separately:
- For the [tex]\(x^3\)[/tex] term: [tex]\(3x^3 + 5x^3 = 8x^3\)[/tex]
- For the [tex]\(x^2\)[/tex] term: There are no [tex]\(x^2\)[/tex] terms, so the sum is [tex]\(0x^2\)[/tex].
- For the [tex]\(x\)[/tex] term: [tex]\(-5x + 7x = 2x\)[/tex]
- For the constant term: [tex]\(-8 + 3 = -5\)[/tex]
4. Combine the results into one polynomial:
[tex]\[
8x^3 + 0x^2 + 2x - 5
\][/tex]
5. Simplify the polynomial by removing the zero coefficient term:
[tex]\[
8x^3 + 2x - 5
\][/tex]
Therefore, the correct sum of [tex]\((3x^3 - 5x - 8) + (5x^3 + 7x + 3)\)[/tex] is [tex]\(8x^3 + 2x - 5\)[/tex].
Hence, the correct answer is:
[tex]\[
8x^3 + 2x - 5
\][/tex]
