Answer :
To determine which polynomial functions are written in standard form, we need to ensure that the terms of each polynomial are ordered by descending powers of [tex]\(x\)[/tex].
Let's examine each polynomial:
1. [tex]\( f(x) = 8 - x^5 \)[/tex]
- The polynomial has terms [tex]\(8\)[/tex] and [tex]\(-x^5\)[/tex].
- Rearranging these terms in descending order by the powers of [tex]\(x\)[/tex], we get [tex]\(-x^5 + 8\)[/tex].
- Therefore, [tex]\(f(x) = 8 - x^5\)[/tex] is not in standard form.
2. [tex]\( f(x) = -3x^5 + 5x - 2 \)[/tex]
- The polynomial has terms [tex]\(-3x^5\)[/tex], [tex]\(5x\)[/tex], and [tex]\(-2\)[/tex].
- These terms are already arranged as [tex]\( -3x^5 \)[/tex] (power of 5), [tex]\( 5x \)[/tex] (power of 1), and [tex]\( -2 \)[/tex] (power of 0).
- This is in descending order by the powers of [tex]\(x\)[/tex].
- Therefore, [tex]\(f(x) = -3x^5 + 5x - 2\)[/tex] is in standard form.
3. [tex]\( f(x) = 2x^5 + 2x + x^3 \)[/tex]
- The polynomial has terms [tex]\(2x^5\)[/tex], [tex]\(2x\)[/tex], and [tex]\(x^3\)[/tex].
- Rearranging these terms in descending order by the powers of [tex]\(x\)[/tex], we get [tex]\(2x^5 + x^3 + 2x\)[/tex].
- Therefore, [tex]\(f(x) = 2x^5 + 2x + x^3\)[/tex] is not in standard form.
4. [tex]\( f(x) = x^3 - 8x^2 \)[/tex]
- The polynomial has terms [tex]\(x^3\)[/tex] and [tex]\(-8x^2\)[/tex].
- These terms are already arranged as [tex]\( x^3 \)[/tex] (power of 3) and [tex]\( -8x^2 \)[/tex] (power of 2).
- This is in descending order by the powers of [tex]\(x\)[/tex].
- Therefore, [tex]\(f(x) = x^3 - 8x^2\)[/tex] is in standard form.
Thus, the polynomial functions written in standard form are:
- [tex]\( f(x) = -3x^5 + 5x - 2 \)[/tex]
- [tex]\( f(x) = x^3 - 8x^2 \)[/tex]
Next, we need to find the degree of the polynomial [tex]\( f(x) = -3x^5 + 4x - 2 \)[/tex]:
The degree of a polynomial is the highest power of [tex]\(x\)[/tex] with a non-zero coefficient.
- The terms in [tex]\( f(x) = -3x^5 + 4x - 2 \)[/tex] are [tex]\(-3x^5\)[/tex] (power of 5), [tex]\(4x\)[/tex] (power of 1), and [tex]\(-2\)[/tex] (power of 0).
- The highest power of [tex]\(x\)[/tex] in this polynomial is [tex]\(5\)[/tex].
Therefore, the degree of the polynomial [tex]\( f(x) = -3x^5 + 4x - 2 \)[/tex] is [tex]\(5\)[/tex].
Summarizing:
1. The polynomial functions written in standard form are:
- [tex]\( f(x) = -3x^5 + 5x - 2 \)[/tex]
- [tex]\( f(x) = x^3 - 8x^2 \)[/tex]
2. The degree of the polynomial [tex]\( f(x) = -3x^5 + 4x - 2 \)[/tex] is [tex]\(5\)[/tex].
Let's examine each polynomial:
1. [tex]\( f(x) = 8 - x^5 \)[/tex]
- The polynomial has terms [tex]\(8\)[/tex] and [tex]\(-x^5\)[/tex].
- Rearranging these terms in descending order by the powers of [tex]\(x\)[/tex], we get [tex]\(-x^5 + 8\)[/tex].
- Therefore, [tex]\(f(x) = 8 - x^5\)[/tex] is not in standard form.
2. [tex]\( f(x) = -3x^5 + 5x - 2 \)[/tex]
- The polynomial has terms [tex]\(-3x^5\)[/tex], [tex]\(5x\)[/tex], and [tex]\(-2\)[/tex].
- These terms are already arranged as [tex]\( -3x^5 \)[/tex] (power of 5), [tex]\( 5x \)[/tex] (power of 1), and [tex]\( -2 \)[/tex] (power of 0).
- This is in descending order by the powers of [tex]\(x\)[/tex].
- Therefore, [tex]\(f(x) = -3x^5 + 5x - 2\)[/tex] is in standard form.
3. [tex]\( f(x) = 2x^5 + 2x + x^3 \)[/tex]
- The polynomial has terms [tex]\(2x^5\)[/tex], [tex]\(2x\)[/tex], and [tex]\(x^3\)[/tex].
- Rearranging these terms in descending order by the powers of [tex]\(x\)[/tex], we get [tex]\(2x^5 + x^3 + 2x\)[/tex].
- Therefore, [tex]\(f(x) = 2x^5 + 2x + x^3\)[/tex] is not in standard form.
4. [tex]\( f(x) = x^3 - 8x^2 \)[/tex]
- The polynomial has terms [tex]\(x^3\)[/tex] and [tex]\(-8x^2\)[/tex].
- These terms are already arranged as [tex]\( x^3 \)[/tex] (power of 3) and [tex]\( -8x^2 \)[/tex] (power of 2).
- This is in descending order by the powers of [tex]\(x\)[/tex].
- Therefore, [tex]\(f(x) = x^3 - 8x^2\)[/tex] is in standard form.
Thus, the polynomial functions written in standard form are:
- [tex]\( f(x) = -3x^5 + 5x - 2 \)[/tex]
- [tex]\( f(x) = x^3 - 8x^2 \)[/tex]
Next, we need to find the degree of the polynomial [tex]\( f(x) = -3x^5 + 4x - 2 \)[/tex]:
The degree of a polynomial is the highest power of [tex]\(x\)[/tex] with a non-zero coefficient.
- The terms in [tex]\( f(x) = -3x^5 + 4x - 2 \)[/tex] are [tex]\(-3x^5\)[/tex] (power of 5), [tex]\(4x\)[/tex] (power of 1), and [tex]\(-2\)[/tex] (power of 0).
- The highest power of [tex]\(x\)[/tex] in this polynomial is [tex]\(5\)[/tex].
Therefore, the degree of the polynomial [tex]\( f(x) = -3x^5 + 4x - 2 \)[/tex] is [tex]\(5\)[/tex].
Summarizing:
1. The polynomial functions written in standard form are:
- [tex]\( f(x) = -3x^5 + 5x - 2 \)[/tex]
- [tex]\( f(x) = x^3 - 8x^2 \)[/tex]
2. The degree of the polynomial [tex]\( f(x) = -3x^5 + 4x - 2 \)[/tex] is [tex]\(5\)[/tex].
