Answer :
We begin by writing the two polynomials clearly. The first polynomial is
[tex]$$
2x^7 + 5x + 4,
$$[/tex]
and the second polynomial is
[tex]$$
5x^9 + 8x.
$$[/tex]
Since we are adding these polynomials, we combine the like terms.
1. For the term with [tex]$x^9$[/tex], there is no [tex]$x^9$[/tex] term in the first polynomial (which we can think of as having a coefficient of 0), and the second polynomial has a coefficient of 5. Thus, the sum for the [tex]$x^9$[/tex] term is:
[tex]$$
0 + 5 = 5,
$$[/tex]
resulting in the term
[tex]$$
5x^9.
$$[/tex]
2. For the term with [tex]$x^7$[/tex], the first polynomial provides a coefficient of 2, while the second polynomial has no [tex]$x^7$[/tex] term. Therefore, we have:
[tex]$$
2 + 0 = 2,
$$[/tex]
which gives
[tex]$$
2x^7.
$$[/tex]
3. For the term with [tex]$x$[/tex], the first polynomial has a coefficient of 5 and the second has 8, so:
[tex]$$
5 + 8 = 13,
$$[/tex]
resulting in
[tex]$$
13x.
$$[/tex]
4. Finally, for the constant term, the first polynomial gives 4 and the second has none, so:
[tex]$$
4 + 0 = 4.
$$[/tex]
Combining all these, the sum of the two polynomials is
[tex]$$
5x^9 + 2x^7 + 13x + 4.
$$[/tex]
Thus, the polynomial representing the sum is
[tex]$$
\boxed{5x^9 + 2x^7 + 13x + 4}.
$$[/tex]
[tex]$$
2x^7 + 5x + 4,
$$[/tex]
and the second polynomial is
[tex]$$
5x^9 + 8x.
$$[/tex]
Since we are adding these polynomials, we combine the like terms.
1. For the term with [tex]$x^9$[/tex], there is no [tex]$x^9$[/tex] term in the first polynomial (which we can think of as having a coefficient of 0), and the second polynomial has a coefficient of 5. Thus, the sum for the [tex]$x^9$[/tex] term is:
[tex]$$
0 + 5 = 5,
$$[/tex]
resulting in the term
[tex]$$
5x^9.
$$[/tex]
2. For the term with [tex]$x^7$[/tex], the first polynomial provides a coefficient of 2, while the second polynomial has no [tex]$x^7$[/tex] term. Therefore, we have:
[tex]$$
2 + 0 = 2,
$$[/tex]
which gives
[tex]$$
2x^7.
$$[/tex]
3. For the term with [tex]$x$[/tex], the first polynomial has a coefficient of 5 and the second has 8, so:
[tex]$$
5 + 8 = 13,
$$[/tex]
resulting in
[tex]$$
13x.
$$[/tex]
4. Finally, for the constant term, the first polynomial gives 4 and the second has none, so:
[tex]$$
4 + 0 = 4.
$$[/tex]
Combining all these, the sum of the two polynomials is
[tex]$$
5x^9 + 2x^7 + 13x + 4.
$$[/tex]
Thus, the polynomial representing the sum is
[tex]$$
\boxed{5x^9 + 2x^7 + 13x + 4}.
$$[/tex]
