To determine which polynomial is in standard form, we need to ensure that the terms in each polynomial are arranged in descending order of their exponents. Let's examine each polynomial step by step:
1. Polynomial 1: [tex]\(12x - 14x^4 + 11x^5\)[/tex]
- The terms are: [tex]\(12x\)[/tex], [tex]\(-14x^4\)[/tex], and [tex]\(11x^5\)[/tex].
- The exponents in descending order should be [tex]\(5, 4, 1\)[/tex].
- In the given form, the order is [tex]\(1, 4, 5\)[/tex], which is not correct.
- Therefore, Polynomial 1 is not in standard form.
2. Polynomial 2: [tex]\(-6x - 3x^2 + 2\)[/tex]
- The terms are: [tex]\(-6x\)[/tex], [tex]\(-3x^2\)[/tex], and [tex]\(2\)[/tex].
- The exponents in descending order should be [tex]\(2, 1, 0\)[/tex].
- In the given form, the order is [tex]\(1, 2, 0\)[/tex], which is not correct.
- Therefore, Polynomial 2 is not in standard form.
3. Polynomial 3: [tex]\(11x^3 - 6x^2 + 5x\)[/tex]
- The terms are: [tex]\(11x^3\)[/tex], [tex]\(-6x^2\)[/tex], and [tex]\(5x\)[/tex].
- The exponents in descending order should be [tex]\(3, 2, 1\)[/tex].
- In the given form, the order is [tex]\(3, 2, 1\)[/tex], which is correct.
- Therefore, Polynomial 3 is in standard form.
4. Polynomial 4: [tex]\(14x^9 + 15x^{12} + 17\)[/tex]
- The terms are: [tex]\(14x^9\)[/tex], [tex]\(15x^{12}\)[/tex], and [tex]\(17\)[/tex].
- The exponents in descending order should be [tex]\(12, 9, 0\)[/tex].
- In the given form, the order is [tex]\(9, 12, 0\)[/tex], which is not correct.
- Therefore, Polynomial 4 is not in standard form.
Conclusion:
Among the given polynomials, only Polynomial 3: [tex]\(11x^3 - 6x^2 + 5x\)[/tex] is in standard form.
