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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 degree (from highest to lowest degree).
Let's examine each of the given polynomials:
1. [tex]\(2x^4 + 6 + 24x^5\)[/tex]
- Current order: [tex]\(2x^4 + 6 + 24x^5\)[/tex]
- Standard form requires the highest degree first. Here, [tex]\(24x^5\)[/tex] is the term with the highest degree, so the correct order is [tex]\(24x^5 + 2x^4 + 6\)[/tex].
2. [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
- Current order: [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
- The degrees of the terms are [tex]\(2\)[/tex], [tex]\(3\)[/tex], and [tex]\(4\)[/tex]. The correct order is [tex]\(12x^4 - 9x^3 + 6x^2\)[/tex].
3. [tex]\(19x + 6x^2 + 2\)[/tex]
- Current order: [tex]\(19x + 6x^2 + 2\)[/tex]
- The degrees are [tex]\(1\)[/tex], [tex]\(2\)[/tex], and [tex]\(0\)[/tex]. The correct order is [tex]\(6x^2 + 19x + 2\)[/tex].
4. [tex]\(23x^9 - 12x^4 + 19\)[/tex]
- Current order: [tex]\(23x^9 - 12x^4 + 19\)[/tex]
- The degrees are [tex]\(9\)[/tex], [tex]\(4\)[/tex], and [tex]\(0\)[/tex], which are already in descending order.
After reviewing each polynomial, we can see that the polynomial already in standard form is:
[tex]\(23x^9 - 12x^4 + 19\)[/tex]
This is because the terms are correctly ordered from highest to lowest degree.
Let's examine each of the given polynomials:
1. [tex]\(2x^4 + 6 + 24x^5\)[/tex]
- Current order: [tex]\(2x^4 + 6 + 24x^5\)[/tex]
- Standard form requires the highest degree first. Here, [tex]\(24x^5\)[/tex] is the term with the highest degree, so the correct order is [tex]\(24x^5 + 2x^4 + 6\)[/tex].
2. [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
- Current order: [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
- The degrees of the terms are [tex]\(2\)[/tex], [tex]\(3\)[/tex], and [tex]\(4\)[/tex]. The correct order is [tex]\(12x^4 - 9x^3 + 6x^2\)[/tex].
3. [tex]\(19x + 6x^2 + 2\)[/tex]
- Current order: [tex]\(19x + 6x^2 + 2\)[/tex]
- The degrees are [tex]\(1\)[/tex], [tex]\(2\)[/tex], and [tex]\(0\)[/tex]. The correct order is [tex]\(6x^2 + 19x + 2\)[/tex].
4. [tex]\(23x^9 - 12x^4 + 19\)[/tex]
- Current order: [tex]\(23x^9 - 12x^4 + 19\)[/tex]
- The degrees are [tex]\(9\)[/tex], [tex]\(4\)[/tex], and [tex]\(0\)[/tex], which are already in descending order.
After reviewing each polynomial, we can see that the polynomial already in standard form is:
[tex]\(23x^9 - 12x^4 + 19\)[/tex]
This is because the terms are correctly ordered from highest to lowest degree.
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