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Answer :
To determine which polynomial is in standard form, we need to ensure that the polynomial is written with its terms in descending order, based on the exponents of the variable [tex]\( x \)[/tex].
Let’s examine each option:
1. Option 1: [tex]\(2x^4 + 6 + 24x^5\)[/tex]
- Look at the exponents: [tex]\(4\)[/tex] in [tex]\(2x^4\)[/tex], [tex]\(5\)[/tex] in [tex]\(24x^5\)[/tex], and the constant [tex]\(6\)[/tex].
- To be in standard form, the term with the highest exponent should come first. Here, [tex]\(24x^5\)[/tex] should be placed before all other terms. This is not in standard form.
2. Option 2: [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
- Look at the exponents: [tex]\(2\)[/tex] in [tex]\(6x^2\)[/tex], [tex]\(3\)[/tex] in [tex]\(-9x^3\)[/tex], and [tex]\(4\)[/tex] in [tex]\(12x^4\)[/tex].
- The highest exponent is [tex]\(4\)[/tex], but the term [tex]\(12x^4\)[/tex] is placed last. This is not in standard form.
3. Option 3: [tex]\(19x + 6x^2 + 2\)[/tex]
- Look at the exponents: [tex]\(1\)[/tex] in [tex]\(19x\)[/tex], [tex]\(2\)[/tex] in [tex]\(6x^2\)[/tex], and the constant [tex]\(2\)[/tex].
- The term with the highest exponent [tex]\(6x^2\)[/tex] should be listed before [tex]\(19x\)[/tex]. This is not in standard form.
4. Option 4: [tex]\(23x^9 - 12x^4 + 19\)[/tex]
- Look at the exponents: [tex]\(9\)[/tex] in [tex]\(23x^9\)[/tex], [tex]\(4\)[/tex] in [tex]\(-12x^4\)[/tex], and the constant [tex]\(19\)[/tex].
- The terms are already arranged in descending order from the highest exponent [tex]\(9\)[/tex] to the lowest, the constant. This is in standard form.
Thus, Option 4: [tex]\(23x^9 - 12x^4 + 19\)[/tex] is the polynomial that is in standard form, with the terms correctly arranged by decreasing powers of [tex]\( x \)[/tex].
Let’s examine each option:
1. Option 1: [tex]\(2x^4 + 6 + 24x^5\)[/tex]
- Look at the exponents: [tex]\(4\)[/tex] in [tex]\(2x^4\)[/tex], [tex]\(5\)[/tex] in [tex]\(24x^5\)[/tex], and the constant [tex]\(6\)[/tex].
- To be in standard form, the term with the highest exponent should come first. Here, [tex]\(24x^5\)[/tex] should be placed before all other terms. This is not in standard form.
2. Option 2: [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
- Look at the exponents: [tex]\(2\)[/tex] in [tex]\(6x^2\)[/tex], [tex]\(3\)[/tex] in [tex]\(-9x^3\)[/tex], and [tex]\(4\)[/tex] in [tex]\(12x^4\)[/tex].
- The highest exponent is [tex]\(4\)[/tex], but the term [tex]\(12x^4\)[/tex] is placed last. This is not in standard form.
3. Option 3: [tex]\(19x + 6x^2 + 2\)[/tex]
- Look at the exponents: [tex]\(1\)[/tex] in [tex]\(19x\)[/tex], [tex]\(2\)[/tex] in [tex]\(6x^2\)[/tex], and the constant [tex]\(2\)[/tex].
- The term with the highest exponent [tex]\(6x^2\)[/tex] should be listed before [tex]\(19x\)[/tex]. This is not in standard form.
4. Option 4: [tex]\(23x^9 - 12x^4 + 19\)[/tex]
- Look at the exponents: [tex]\(9\)[/tex] in [tex]\(23x^9\)[/tex], [tex]\(4\)[/tex] in [tex]\(-12x^4\)[/tex], and the constant [tex]\(19\)[/tex].
- The terms are already arranged in descending order from the highest exponent [tex]\(9\)[/tex] to the lowest, the constant. This is in standard form.
Thus, Option 4: [tex]\(23x^9 - 12x^4 + 19\)[/tex] is the polynomial that is in standard form, with the terms correctly arranged by decreasing powers of [tex]\( x \)[/tex].
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