To determine which polynomials are in standard form, we need to understand what standard form means for a polynomial:
A polynomial is said to be in standard form if its terms are written in descending order of their degree. The degree of a term is determined by the exponent of the variable in that term.
Let's evaluate each option:
A. [tex]\(3z - 1\)[/tex]:
- The term [tex]\(3z\)[/tex] is [tex]\(z^1\)[/tex].
- The term [tex]\(-1\)[/tex] is [tex]\(z^0\)[/tex].
- The terms are in order of descending degrees: [tex]\(z^1\)[/tex] and [tex]\(z^0\)[/tex].
- Therefore, this polynomial is in standard form.
B. [tex]\(2 + 4x - 5x^2\)[/tex]:
- The term [tex]\(2\)[/tex] is [tex]\(x^0\)[/tex].
- The term [tex]\(4x\)[/tex] is [tex]\(x^1\)[/tex].
- The term [tex]\(-5x^2\)[/tex] is [tex]\(x^2\)[/tex].
- The terms are not in order of descending degrees; they should be ordered as [tex]\(-5x^2 + 4x + 2\)[/tex].
- Therefore, this polynomial is not in standard form.
C. [tex]\(-5p^5 + 2p^2 - 3p + 1\)[/tex]:
- The term [tex]\(-5p^5\)[/tex] is [tex]\(p^5\)[/tex].
- The term [tex]\(2p^2\)[/tex] is [tex]\(p^2\)[/tex].
- The term [tex]\(-3p\)[/tex] is [tex]\(p^1\)[/tex].
- The term [tex]\(1\)[/tex] is [tex]\(p^0\)[/tex].
- The terms are in order of descending degrees: [tex]\(p^5, p^2, p^1,\)[/tex] and [tex]\(p^0\)[/tex].
- Therefore, this polynomial is in standard form.
D. None of the above:
- This option is not correct because there are valid polynomials in standard form in options A and C.
So, the polynomials that are in standard form are:
- Option A: [tex]\(3z - 1\)[/tex]
- Option C: [tex]\(-5p^5 + 2p^2 - 3p + 1\)[/tex]
Thus, the correct answers are:
A. [tex]\(3z - 1\)[/tex]
C. [tex]\(-5p^5 + 2p^2 - 3p + 1\)[/tex]
Therefore, the answer is:
1, 3
