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Answer :
To determine which polynomial is in standard form, we need to understand what "standard form" means for a polynomial. A polynomial is in standard form when its terms are written in descending order of their exponents.
Let's examine each polynomial given in the question:
1. [tex]\( 2x^4 + 6 + 24x^5 \)[/tex]
We need to rewrite this polynomial in descending order of the exponents:
[tex]\[
24x^5 + 2x^4 + 6
\][/tex]
This is now in standard form, but it wasn't originally given in standard form.
2. [tex]\( 6x^2 - 9x^3 + 12x^4 \)[/tex]
Rewrite this polynomial in descending order of the exponents:
[tex]\[
12x^4 - 9x^3 + 6x^2
\][/tex]
This is now in standard form, but it wasn't originally given in standard form.
3. [tex]\( 19x + 6x^2 + 2 \)[/tex]
Rewrite this polynomial in descending order of the exponents:
[tex]\[
6x^2 + 19x + 2
\][/tex]
This is now in standard form, but it wasn't originally given in standard form.
4. [tex]\( 23x^9 - 12x^4 + 19 \)[/tex]
This polynomial is already written in descending order of the exponents:
[tex]\[
23x^9 - 12x^4 + 19
\][/tex]
This means it is already in standard form.
Thus, the only polynomial that is already in standard form is:
[tex]\( 23x^9 - 12x^4 + 19 \)[/tex].
So, the answer is:
[tex]\( 23x^9 - 12x^4 + 19 \)[/tex].
Let's examine each polynomial given in the question:
1. [tex]\( 2x^4 + 6 + 24x^5 \)[/tex]
We need to rewrite this polynomial in descending order of the exponents:
[tex]\[
24x^5 + 2x^4 + 6
\][/tex]
This is now in standard form, but it wasn't originally given in standard form.
2. [tex]\( 6x^2 - 9x^3 + 12x^4 \)[/tex]
Rewrite this polynomial in descending order of the exponents:
[tex]\[
12x^4 - 9x^3 + 6x^2
\][/tex]
This is now in standard form, but it wasn't originally given in standard form.
3. [tex]\( 19x + 6x^2 + 2 \)[/tex]
Rewrite this polynomial in descending order of the exponents:
[tex]\[
6x^2 + 19x + 2
\][/tex]
This is now in standard form, but it wasn't originally given in standard form.
4. [tex]\( 23x^9 - 12x^4 + 19 \)[/tex]
This polynomial is already written in descending order of the exponents:
[tex]\[
23x^9 - 12x^4 + 19
\][/tex]
This means it is already in standard form.
Thus, the only polynomial that is already in standard form is:
[tex]\( 23x^9 - 12x^4 + 19 \)[/tex].
So, the answer is:
[tex]\( 23x^9 - 12x^4 + 19 \)[/tex].
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