College

Which polynomial represents the sum below?

[tex]
\[
\begin{array}{r}
2x^7 + 5x + 4 \\
+\quad 5x^9 + 8x \\
\hline
\end{array}
\]
[/tex]

A. [tex]\(7x^9 + 13x + 4\)[/tex]

B. [tex]\(7x^{16} + 13x + 4\)[/tex]

C. [tex]\(5x^9 + 7x^7 + 13x + 4\)[/tex]

D. [tex]\(5x^9 + 2x^7 + 13x + 4\)[/tex]

Answer :

Let's add the polynomials step-by-step.

We are given the expression:
[tex]\[
\begin{array}{r}
2x^7 + 5x + 4 \\
+\quad 5x^9 + 8x \\
\hline
\end{array}
\][/tex]

To add these polynomials, we combine like terms, which are terms with the same power of [tex]\(x\)[/tex].

Let's identify and combine each set of like terms:

1. Terms with [tex]\(x^9\)[/tex]:
- The only [tex]\(x^9\)[/tex] term is from the second polynomial: [tex]\(5x^9\)[/tex].

2. Terms with [tex]\(x^7\)[/tex]:
- The only [tex]\(x^7\)[/tex] term is from the first polynomial: [tex]\(2x^7\)[/tex].

3. Terms with [tex]\(x\)[/tex]:
- From the first polynomial, we have [tex]\(5x\)[/tex].
- From the second polynomial, we have [tex]\(8x\)[/tex].
- Adding these gives us [tex]\(5x + 8x = 13x\)[/tex].

4. Constant terms:
- The constant term from the first polynomial is [tex]\(4\)[/tex].
- There is no constant term in the second polynomial.
- Therefore, the constant term in the sum is [tex]\(4\)[/tex].

Putting all these together, the sum of the polynomials is:
[tex]\[ 5x^9 + 2x^7 + 13x + 4 \][/tex]

Thus, the polynomial that represents the sum is:

D. [tex]\(5x^9 + 2x^7 + 13x + 4\)[/tex]