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Answer :
To determine which polynomial is in standard form, let's first understand what standard form means for polynomials. A polynomial is in standard form when its terms are ordered in descending order of their exponents.
Let's analyze each given polynomial:
1. [tex]\(2x^4 + 6 + 24x^5\)[/tex]
- The terms are [tex]\(24x^5\)[/tex], [tex]\(2x^4\)[/tex], and [tex]\(6\)[/tex].
- The exponents are: [tex]\(5\)[/tex] (for [tex]\(24x^5\)[/tex]), [tex]\(4\)[/tex] (for [tex]\(2x^4\)[/tex]), and [tex]\(0\)[/tex] (for [tex]\(6\)[/tex]).
- The correct order should be [tex]\(24x^5 + 2x^4 + 6\)[/tex]. Therefore, this polynomial is not in standard form.
2. [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
- The terms are [tex]\(12x^4\)[/tex], [tex]\(-9x^3\)[/tex], and [tex]\(6x^2\)[/tex].
- The exponents are: [tex]\(4\)[/tex] (for [tex]\(12x^4\)[/tex]), [tex]\(3\)[/tex] (for [tex]\(-9x^3\)[/tex]), and [tex]\(2\)[/tex] (for [tex]\(6x^2\)[/tex]).
- The correct order should be [tex]\(12x^4 - 9x^3 + 6x^2\)[/tex]. Therefore, this polynomial is not in standard form.
3. [tex]\(19x + 6x^2 + 2\)[/tex]
- The terms are [tex]\(6x^2\)[/tex], [tex]\(19x\)[/tex], and [tex]\(2\)[/tex].
- The exponents are: [tex]\(2\)[/tex] (for [tex]\(6x^2\)[/tex]), [tex]\(1\)[/tex] (for [tex]\(19x\)[/tex]), and [tex]\(0\)[/tex] (for [tex]\(2\)[/tex]).
- The correct order should be [tex]\(6x^2 + 19x + 2\)[/tex]. Therefore, this polynomial is not in standard form.
4. [tex]\(23x^9 - 12x^4 + 19\)[/tex]
- The terms are [tex]\(23x^9\)[/tex], [tex]\(-12x^4\)[/tex], and [tex]\(19\)[/tex].
- The exponents are: [tex]\(9\)[/tex] (for [tex]\(23x^9\)[/tex]), [tex]\(4\)[/tex] (for [tex]\(-12x^4\)[/tex]), and [tex]\(0\)[/tex] (for [tex]\(19\)[/tex]).
- The correct order is [tex]\(23x^9 - 12x^4 + 19\)[/tex]. Therefore, this polynomial is in standard form.
Thus, the polynomial in standard form is:
[tex]\[ \boxed{23x^9 - 12x^4 + 19} \][/tex]
Let's analyze each given polynomial:
1. [tex]\(2x^4 + 6 + 24x^5\)[/tex]
- The terms are [tex]\(24x^5\)[/tex], [tex]\(2x^4\)[/tex], and [tex]\(6\)[/tex].
- The exponents are: [tex]\(5\)[/tex] (for [tex]\(24x^5\)[/tex]), [tex]\(4\)[/tex] (for [tex]\(2x^4\)[/tex]), and [tex]\(0\)[/tex] (for [tex]\(6\)[/tex]).
- The correct order should be [tex]\(24x^5 + 2x^4 + 6\)[/tex]. Therefore, this polynomial is not in standard form.
2. [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
- The terms are [tex]\(12x^4\)[/tex], [tex]\(-9x^3\)[/tex], and [tex]\(6x^2\)[/tex].
- The exponents are: [tex]\(4\)[/tex] (for [tex]\(12x^4\)[/tex]), [tex]\(3\)[/tex] (for [tex]\(-9x^3\)[/tex]), and [tex]\(2\)[/tex] (for [tex]\(6x^2\)[/tex]).
- The correct order should be [tex]\(12x^4 - 9x^3 + 6x^2\)[/tex]. Therefore, this polynomial is not in standard form.
3. [tex]\(19x + 6x^2 + 2\)[/tex]
- The terms are [tex]\(6x^2\)[/tex], [tex]\(19x\)[/tex], and [tex]\(2\)[/tex].
- The exponents are: [tex]\(2\)[/tex] (for [tex]\(6x^2\)[/tex]), [tex]\(1\)[/tex] (for [tex]\(19x\)[/tex]), and [tex]\(0\)[/tex] (for [tex]\(2\)[/tex]).
- The correct order should be [tex]\(6x^2 + 19x + 2\)[/tex]. Therefore, this polynomial is not in standard form.
4. [tex]\(23x^9 - 12x^4 + 19\)[/tex]
- The terms are [tex]\(23x^9\)[/tex], [tex]\(-12x^4\)[/tex], and [tex]\(19\)[/tex].
- The exponents are: [tex]\(9\)[/tex] (for [tex]\(23x^9\)[/tex]), [tex]\(4\)[/tex] (for [tex]\(-12x^4\)[/tex]), and [tex]\(0\)[/tex] (for [tex]\(19\)[/tex]).
- The correct order is [tex]\(23x^9 - 12x^4 + 19\)[/tex]. Therefore, this polynomial is in standard form.
Thus, the polynomial in standard form is:
[tex]\[ \boxed{23x^9 - 12x^4 + 19} \][/tex]
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