To determine which polynomial is in standard form, we must ensure that the terms in the polynomial are arranged in descending order of their degrees.
Let's examine each polynomial one by one:
1. Polynomial 1: \(1 + 2x - 8x^2 + 6x^3\)
- Degree of the terms: \(6x^3\) (degree 3), \(-8x^2\) (degree 2), \(2x\) (degree 1), \(1\) (degree 0).
- The terms are not arranged in descending order, as the term \(6x^3\) should come first.
- Therefore, this polynomial is not in standard form.
2. Polynomial 2: \(2x^2 + 6x^3 - 9x + 12\)
- Degree of the terms: \(6x^3\) (degree 3), \(2x^2\) (degree 2), \(-9x\) (degree 1), \(12\) (degree 0).
- The terms are not arranged in descending order, as the term \(6x^3\) should come first.
- Therefore, this polynomial is not in standard form.
3. Polynomial 3: \(6x^3 + 5x - 3x^2 + 2\)
- Degree of the terms: \(6x^3\) (degree 3), \(-3x^2\) (degree 2), \(5x\) (degree 1), \(2\) (degree 0).
- The terms are arranged in descending order: \(6x^3\), \(-3x^2\), \(5x\), \(2\).
- Therefore, this polynomial is in standard form.
4. Polynomial 4: \(2x^3 + 4x^2 - 7x + 5\)
- Degree of the terms: \(2x^3\) (degree 3), \(4x^2\) (degree 2), \(-7x\) (degree 1), \(5\) (degree 0).
- The terms are arranged in descending order: \(2x^3\), \(4x^2\), \(-7x\), \(5\).
- Therefore, this polynomial is in standard form.
Thus, the polynomials in standard form are:
- Polynomial 3: \(6x^3 + 5x - 3x^2 + 2\)
- Polynomial 4: \(2x^3 + 4x^2 - 7x + 5\)
By listing their positions, the respective results are:
- \(1\) (not in standard form)
- \(2\) (not in standard form)
- \(3\) (in standard form)
- \(4\) (in standard form)
So the answer is:
[tex]\[
(0, 0, 3, 4)
\][/tex]
