To determine which of the given polynomials are in standard form, we need to check if each polynomial is written with its terms ordered from the highest degree to the lowest degree. Let's break down each polynomial:
1. Polynomial: [tex]\(x^2 + 3x + 2\)[/tex]
- The terms are [tex]\(x^2\)[/tex], [tex]\(3x\)[/tex], and [tex]\(2\)[/tex].
- The degrees are 2, 1, and 0, respectively.
- These terms are already ordered by decreasing degree.
- Therefore, [tex]\(x^2 + 3x + 2\)[/tex] is in standard form.
2. Polynomial: [tex]\(q^3 - 15q + 12q^2 - 16\)[/tex]
- The terms are [tex]\(q^3\)[/tex], [tex]\(-15q\)[/tex], [tex]\(12q^2\)[/tex] and [tex]\(-16\)[/tex].
- The degrees are 3, 1, 2, and 0, respectively.
- These terms are not ordered by decreasing degree.
- Therefore, [tex]\(q^3 - 15q + 12q^2 - 16\)[/tex] is not in standard form. The correct standard form would be [tex]\(q^3 + 12q^2 - 15q - 16\)[/tex].
3. Polynomial: [tex]\(4a + a^2 + a - 2\)[/tex]
- The terms are [tex]\(4a\)[/tex], [tex]\(a^2\)[/tex], [tex]\(a\)[/tex], and [tex]\(-2\)[/tex].
- The degrees are 1, 2, 1, and 0, respectively.
- These terms are not ordered by decreasing degree.
- Therefore, [tex]\(4a + a^2 + a - 2\)[/tex] is not in standard form. The correct standard form would be [tex]\(a^2 + 5a - 2\)[/tex].
4. Polynomial: [tex]\(3x^4 + 4x^3 - 3x^2 - 1\)[/tex]
- The terms are [tex]\(3x^4\)[/tex], [tex]\(4x^3\)[/tex], [tex]\(-3x^2\)[/tex], and [tex]\(-1\)[/tex].
- The degrees are 4, 3, 2, and 0, respectively.
- These terms are already ordered by decreasing degree.
- Therefore, [tex]\(3x^4 + 4x^3 - 3x^2 - 1\)[/tex] is in standard form.
5. Polynomial: [tex]\(3t^3 + 3t^2 + 2t\)[/tex]
- The terms are [tex]\(3t^3\)[/tex], [tex]\(3t^2\)[/tex], and [tex]\(2t\)[/tex].
- The degrees are 3, 2, and 1, respectively.
- These terms are already ordered by decreasing degree.
- Therefore, [tex]\(3t^3 + 3t^2 + 2t\)[/tex] is in standard form.
6. Polynomial: [tex]\(14 + a^3 - 6a + 8a^2\)[/tex]
- The terms are [tex]\(14\)[/tex], [tex]\(a^3\)[/tex], [tex]\(-6a\)[/tex], and [tex]\(8a^2\)[/tex].
- The degrees are 0, 3, 1, and 2, respectively.
- These terms are not ordered by decreasing degree.
- Therefore, [tex]\(14 + a^3 - 6a + 8a^2\)[/tex] is not in standard form. The correct standard form would be [tex]\(a^3 + 8a^2 - 6a + 14\)[/tex].
To summarize, the polynomials that are in standard form are:
- [tex]\(x^2 + 3x + 2\)[/tex]
- [tex]\(3x^4 + 4x^3 - 3x^2 - 1\)[/tex]
- [tex]\(3t^3 + 3t^2 + 2t\)[/tex]
