To determine which polynomial is written in standard form, we must ensure that each polynomial's terms are ordered by decreasing degree of the variable [tex]\( x \)[/tex], starting with the highest degree term. Let's analyze each polynomial one by one.
1. First Polynomial: [tex]\( 9 + 2x - 8x^4 + 16x^5 \)[/tex]
Reordering the terms by decreasing degree of [tex]\( x \)[/tex]:
[tex]\[
16x^5 - 8x^4 + 2x + 9
\][/tex]
This polynomial is now in standard form.
2. Second Polynomial: [tex]\( 12x^5 - 6x^2 - 9x + 12 \)[/tex]
Reordering the terms:
[tex]\[
12x^5 - 6x^2 - 9x + 12
\][/tex]
This polynomial is already in standard form.
3. Third Polynomial: [tex]\( 13x^5 + 11x - 6x^2 + 5 \)[/tex]
Reordering the terms:
[tex]\[
13x^5 - 6x^2 + 11x + 5
\][/tex]
This arrangement shows the polynomial in standard form.
4. Fourth Polynomial: [tex]\( 7x^7 + 14x^9 - 17x + 25 \)[/tex]
Reordering the terms:
[tex]\[
14x^9 + 7x^7 - 17x + 25
\][/tex]
This polynomial, when reordered, becomes:
[tex]\[
14x^9 + 7x^7 - 17x + 25
\][/tex]
This is already in standard form.
We have identified and reordered each polynomial. Let’s double-check the order of degrees:
- For the first polynomial, the reordered form is [tex]\( 16x^5 - 8x^4 + 2x + 9 \)[/tex] which is in standard form.
- For the second polynomial, [tex]\( 12x^5 - 6x^2 - 9x + 12 \)[/tex] is already in standard form.
- For the third polynomial, [tex]\( 13x^5 - 6x^2 + 11x + 5 \)[/tex] is in standard form.
- For the fourth polynomial, [tex]\( 14x^9 + 7x^7 - 17x + 25 \)[/tex] is already in standard form.
Note that each polynomial has been confirmed to be in standard form upon inspection. However, the task is to identify the polynomial that was originally presented in standard form without requiring reordering:
- The polynomials that are initially in standard form (without reordering) are:
1. Second Polynomial: [tex]\( 12x^5 - 6x^2 - 9x + 12 \)[/tex]
2. Fourth Polynomial: [tex]\( 7x^7 + 14x^9 - 17x + 25 \)[/tex], which is rewritten to [tex]\( 14x^9 + 7x^7 - 17x + 25 \)[/tex], hence another polynomial that was not in standard form originally.
Therefore, the polynomial [tex]\( 12x^5 - 6x^2 - 9x + 12 \)[/tex] is the one initially presented in standard form without needing to be reordered.
Hence, the correct polynomial in standard form is:
[tex]\[
\boxed{12x^5 - 6x^2 - 9x + 12}
\][/tex]
