Answer :
To determine which polynomial is in standard form, we need to check each polynomial to see if its terms are arranged in order of decreasing exponents, from the highest degree to the lowest degree. Let's review each given polynomial:
1. [tex]$2x^4 + 6 + 24x^5$[/tex]
- Let's arrange the terms by the power of [tex]\(x\)[/tex]: [tex]\(24x^5 + 2x^4 + 6\)[/tex].
- In this arrangement, the terms go from the highest degree (5) to the lowest (constant term 6), so this polynomial is not initially in standard form.
2. [tex]$6x^2 - 9x^3 + 12x^4$[/tex]
- Let's arrange the terms by the power of [tex]\(x\)[/tex]: [tex]\(12x^4 - 9x^3 + 6x^2\)[/tex].
- In this arrangement, the terms go from the highest degree (4) to the lowest (2), so this polynomial is not initially in standard form.
3. [tex]$19x + 6x^2 + 2$[/tex]
- Let's arrange the terms by the power of [tex]\(x\)[/tex]: [tex]\(6x^2 + 19x + 2\)[/tex].
- In this arrangement, the terms go from the highest degree (2) to the lowest (constant term 2), so this polynomial is not initially in standard form.
4. [tex]$23x^9 - 12x^4 + 19$[/tex]
- The terms are already ordered from highest degree to lowest degree: [tex]\(23x^9 - 12x^4 + 19\)[/tex].
- This polynomial is in standard form.
Therefore, the polynomial that is already in standard form is option 4: [tex]\(23x^9 - 12x^4 + 19\)[/tex].
1. [tex]$2x^4 + 6 + 24x^5$[/tex]
- Let's arrange the terms by the power of [tex]\(x\)[/tex]: [tex]\(24x^5 + 2x^4 + 6\)[/tex].
- In this arrangement, the terms go from the highest degree (5) to the lowest (constant term 6), so this polynomial is not initially in standard form.
2. [tex]$6x^2 - 9x^3 + 12x^4$[/tex]
- Let's arrange the terms by the power of [tex]\(x\)[/tex]: [tex]\(12x^4 - 9x^3 + 6x^2\)[/tex].
- In this arrangement, the terms go from the highest degree (4) to the lowest (2), so this polynomial is not initially in standard form.
3. [tex]$19x + 6x^2 + 2$[/tex]
- Let's arrange the terms by the power of [tex]\(x\)[/tex]: [tex]\(6x^2 + 19x + 2\)[/tex].
- In this arrangement, the terms go from the highest degree (2) to the lowest (constant term 2), so this polynomial is not initially in standard form.
4. [tex]$23x^9 - 12x^4 + 19$[/tex]
- The terms are already ordered from highest degree to lowest degree: [tex]\(23x^9 - 12x^4 + 19\)[/tex].
- This polynomial is in standard form.
Therefore, the polynomial that is already in standard form is option 4: [tex]\(23x^9 - 12x^4 + 19\)[/tex].
