Answer :
To determine which polynomial is in standard form, we need to analyze the given options. A polynomial is in standard form when its terms are written in decreasing order of degree, from the highest degree term to the lowest degree term. Let's look at each option:
1. Option 1: [tex]\(2x^4 + 6 + 24x^5\)[/tex]
The terms should be arranged in order of decreasing degree. Here, [tex]\(24x^5\)[/tex] is the highest degree term followed by [tex]\(2x^4\)[/tex], and then the constant term [tex]\(6\)[/tex]. So, the standard form should be:
[tex]\[
24x^5 + 2x^4 + 6
\][/tex]
2. Option 2: [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
The terms here need to be reordered in decreasing order of their degrees. The correct order should be:
[tex]\[
12x^4 - 9x^3 + 6x^2
\][/tex]
3. Option 3: [tex]\(19x + 6x^2 + 2\)[/tex]
Here, [tex]\(6x^2\)[/tex] has the highest degree, followed by [tex]\(19x\)[/tex], then the constant [tex]\(2\)[/tex]. The standard form should be:
[tex]\[
6x^2 + 19x + 2
\][/tex]
4. Option 4: [tex]\(23x^3 - 12x^4 + 19\)[/tex]
This should be reordered to place [tex]\(-12x^4\)[/tex] first, followed by [tex]\(23x^3\)[/tex], and finally the constant [tex]\(19\)[/tex]. So, the standard form is:
[tex]\[
-12x^4 + 23x^3 + 19
\][/tex]
After reorganizing each polynomial, we see none of the original options were in standard form. Now they are properly ordered:
- Option 1 in standard form: [tex]\(24x^5 + 2x^4 + 6\)[/tex]
- Option 2 in standard form: [tex]\(12x^4 - 9x^3 + 6x^2\)[/tex]
- Option 3 in standard form: [tex]\(6x^2 + 19x + 2\)[/tex]
- Option 4 in standard form: [tex]\(-12x^4 + 23x^3 + 19\)[/tex]
1. Option 1: [tex]\(2x^4 + 6 + 24x^5\)[/tex]
The terms should be arranged in order of decreasing degree. Here, [tex]\(24x^5\)[/tex] is the highest degree term followed by [tex]\(2x^4\)[/tex], and then the constant term [tex]\(6\)[/tex]. So, the standard form should be:
[tex]\[
24x^5 + 2x^4 + 6
\][/tex]
2. Option 2: [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
The terms here need to be reordered in decreasing order of their degrees. The correct order should be:
[tex]\[
12x^4 - 9x^3 + 6x^2
\][/tex]
3. Option 3: [tex]\(19x + 6x^2 + 2\)[/tex]
Here, [tex]\(6x^2\)[/tex] has the highest degree, followed by [tex]\(19x\)[/tex], then the constant [tex]\(2\)[/tex]. The standard form should be:
[tex]\[
6x^2 + 19x + 2
\][/tex]
4. Option 4: [tex]\(23x^3 - 12x^4 + 19\)[/tex]
This should be reordered to place [tex]\(-12x^4\)[/tex] first, followed by [tex]\(23x^3\)[/tex], and finally the constant [tex]\(19\)[/tex]. So, the standard form is:
[tex]\[
-12x^4 + 23x^3 + 19
\][/tex]
After reorganizing each polynomial, we see none of the original options were in standard form. Now they are properly ordered:
- Option 1 in standard form: [tex]\(24x^5 + 2x^4 + 6\)[/tex]
- Option 2 in standard form: [tex]\(12x^4 - 9x^3 + 6x^2\)[/tex]
- Option 3 in standard form: [tex]\(6x^2 + 19x + 2\)[/tex]
- Option 4 in standard form: [tex]\(-12x^4 + 23x^3 + 19\)[/tex]
