To determine which polynomial is in standard form, we need to check if the terms are arranged in decreasing order of their degrees.
Let's analyze each polynomial:
1. [tex]\( 12x - 14x^4 + 11x^5 \)[/tex]
- The terms are [tex]\( 12x \)[/tex] (degree 1), [tex]\( -14x^4 \)[/tex] (degree 4), and [tex]\( 11x^5 \)[/tex] (degree 5).
- Their degrees are [tex]\( 1, 4, \)[/tex] and [tex]\( 5 \)[/tex].
- In standard form, the degrees should be in descending order. These terms are not in descending order since [tex]\( 1 < 4 < 5 \)[/tex].
2. [tex]\( -6x - 3x^2 + 2 \)[/tex]
- The terms are [tex]\( -6x \)[/tex] (degree 1), [tex]\( -3x^2 \)[/tex] (degree 2), and [tex]\( 2 \)[/tex] (degree 0).
- Their degrees are [tex]\( 1, 2, \)[/tex] and [tex]\( 0 \)[/tex].
- To be in descending order, we should have [tex]\( 2, 1, \)[/tex] and [tex]\( 0 \)[/tex]. This polynomial is not in descending order since [tex]\( 1, 2, \)[/tex] and [tex]\( 0 \)[/tex] do not follow this order.
3. [tex]\( 11x^3 - 6x^2 + 5x \)[/tex]
- The terms are [tex]\( 11x^3 \)[/tex] (degree 3), [tex]\( -6x^2 \)[/tex] (degree 2), and [tex]\( 5x \)[/tex] (degree 1).
- Their degrees are [tex]\( 3, 2, \)[/tex] and [tex]\( 1 \)[/tex].
- These degrees are indeed in descending order [tex]\( 3 > 2 > 1 \)[/tex]. So this polynomial is in standard form.
4. [tex]\( 14x^9 + 15x^{12} + 17 \)[/tex]
- The terms are [tex]\( 14x^9 \)[/tex] (degree 9), [tex]\( 15x^{12} \)[/tex] (degree 12), and [tex]\( 17 \)[/tex] (degree 0).
- Their degrees are [tex]\( 9, 12, \)[/tex] and [tex]\( 0 \)[/tex].
- To be in standard form, the degrees should be [tex]\( 12 > 9 > 0 \)[/tex]. These terms do not follow this order since [tex]\( 9 < 12 \)[/tex].
Among the given polynomials, the only one that meets the criteria of being in standard form (arranged in descending order of degrees) is option 3, [tex]\( 11x^3 - 6x^2 + 5x \)[/tex].
Thus, the polynomial in standard form is the 3rd polynomial.
