To determine which polynomial is in standard form, we need to ensure that the terms in the polynomial are ordered by descending powers of [tex]\( x \)[/tex]. The highest power (degree) should be listed first, followed by the next highest, and so on, down to the lowest power (constant term, if present).
Let's analyze each polynomial step-by-step:
1. Polynomial: [tex]\( 12x - 14x^4 + 11x^5 \)[/tex]
- Terms: [tex]\( 12x \)[/tex], [tex]\( -14x^4 \)[/tex], [tex]\( 11x^5 \)[/tex]
- Degrees: 1, 4, 5
- Ordered: No (5 > 4 > 1, so it should be [tex]\( 11x^5 - 14x^4 + 12x \)[/tex])
2. Polynomial: [tex]\( -6x - 3x^2 + 2 \)[/tex]
- Terms: [tex]\( -6x \)[/tex], [tex]\( -3x^2 \)[/tex], [tex]\( 2 \)[/tex]
- Degrees: 1, 2, 0
- Ordered: No (2 > 1 > 0, so it should be [tex]\( -3x^2 - 6x + 2 \)[/tex])
3. Polynomial: [tex]\( 11x^3 - 6x^2 + 5x \)[/tex]
- Terms: [tex]\( 11x^3 \)[/tex], [tex]\( -6x^2 \)[/tex], [tex]\( 5x \)[/tex]
- Degrees: 3, 2, 1
- Ordered: Yes (3 > 2 > 1, so it is already in standard form)
4. Polynomial: [tex]\( 14x^9 + 15x^{12} + 17 \)[/tex]
- Terms: [tex]\( 14x^9 \)[/tex], [tex]\( 15x^{12} \)[/tex], [tex]\( 17 \)[/tex]
- Degrees: 9, 12, 0
- Ordered: No (12 > 9 > 0, so it should be [tex]\( 15x^{12} + 14x^9 + 17 \)[/tex])
From this analysis, we can see that only the polynomial [tex]\( 11x^3 - 6x^2 + 5x \)[/tex] is in standard form. Thus, the correct polynomial in standard form is:
[tex]\[ 11x^3 - 6x^2 + 5x \][/tex]
