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Answer :
To determine if a polynomial is in standard form, we must ensure that its terms are arranged in descending order according to their degrees (the exponents on the variable). Let's analyze each option:
1. Consider the polynomial
[tex]$$2x^4 + 6 + 24x^5.$$[/tex]
The degrees of the terms are 4 (for [tex]\(2x^4\)[/tex]), 0 (for [tex]\(6\)[/tex]), and 5 (for [tex]\(24x^5\)[/tex]). In descending order, the term with the highest degree, [tex]\(24x^5\)[/tex] (degree 5), should come first, followed by [tex]\(2x^4\)[/tex] (degree 4), and finally [tex]\(6\)[/tex] (degree 0). Since the terms are not written in this descending order here, this polynomial is not in standard form.
2. Next, look at the polynomial
[tex]$$6x^2 - 9x^3 + 12x^4.$$[/tex]
The degrees of the terms are 2 (for [tex]\(6x^2\)[/tex]), 3 (for [tex]\(-9x^3\)[/tex]), and 4 (for [tex]\(12x^4\)[/tex]). The correct descending order would place [tex]\(12x^4\)[/tex] first, then [tex]\(-9x^3\)[/tex], and finally [tex]\(6x^2\)[/tex]. Because the given order does not follow this sequence, this polynomial is not in standard form.
3. Now, consider the polynomial
[tex]$$19x + 6x^2 + 2.$$[/tex]
The degrees are 1 (for [tex]\(19x\)[/tex]), 2 (for [tex]\(6x^2\)[/tex]), and 0 (for [tex]\(2\)[/tex]). The proper descending order should be [tex]\(6x^2\)[/tex] first, then [tex]\(19x\)[/tex], and finally [tex]\(2\)[/tex]. Since the terms are not in descending order, this polynomial is also not in standard form.
4. Finally, we have the polynomial
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
The degrees here are 9 (for [tex]\(23x^9\)[/tex]), 4 (for [tex]\(-12x^4\)[/tex]), and 0 (for [tex]\(19\)[/tex]). These terms are arranged in descending order (9, then 4, then 0), which means this polynomial is already in standard form.
Based on the analysis, the polynomial in standard form is:
[tex]$$\boxed{23x^9 - 12x^4 + 19}.$$[/tex]
1. Consider the polynomial
[tex]$$2x^4 + 6 + 24x^5.$$[/tex]
The degrees of the terms are 4 (for [tex]\(2x^4\)[/tex]), 0 (for [tex]\(6\)[/tex]), and 5 (for [tex]\(24x^5\)[/tex]). In descending order, the term with the highest degree, [tex]\(24x^5\)[/tex] (degree 5), should come first, followed by [tex]\(2x^4\)[/tex] (degree 4), and finally [tex]\(6\)[/tex] (degree 0). Since the terms are not written in this descending order here, this polynomial is not in standard form.
2. Next, look at the polynomial
[tex]$$6x^2 - 9x^3 + 12x^4.$$[/tex]
The degrees of the terms are 2 (for [tex]\(6x^2\)[/tex]), 3 (for [tex]\(-9x^3\)[/tex]), and 4 (for [tex]\(12x^4\)[/tex]). The correct descending order would place [tex]\(12x^4\)[/tex] first, then [tex]\(-9x^3\)[/tex], and finally [tex]\(6x^2\)[/tex]. Because the given order does not follow this sequence, this polynomial is not in standard form.
3. Now, consider the polynomial
[tex]$$19x + 6x^2 + 2.$$[/tex]
The degrees are 1 (for [tex]\(19x\)[/tex]), 2 (for [tex]\(6x^2\)[/tex]), and 0 (for [tex]\(2\)[/tex]). The proper descending order should be [tex]\(6x^2\)[/tex] first, then [tex]\(19x\)[/tex], and finally [tex]\(2\)[/tex]. Since the terms are not in descending order, this polynomial is also not in standard form.
4. Finally, we have the polynomial
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
The degrees here are 9 (for [tex]\(23x^9\)[/tex]), 4 (for [tex]\(-12x^4\)[/tex]), and 0 (for [tex]\(19\)[/tex]). These terms are arranged in descending order (9, then 4, then 0), which means this polynomial is already in standard form.
Based on the analysis, the polynomial in standard form is:
[tex]$$\boxed{23x^9 - 12x^4 + 19}.$$[/tex]
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