Answer :
To determine which statement best describes the polynomial [tex]\( -24x^7 - 12x^2 - 9x + 6 \)[/tex], we need to analyze the properties and structure of the polynomial.
### Step-by-Step Analysis:
1. Examine the Exponents:
A polynomial is said to be in standard form if its terms are arranged in descending order of the exponents. Let's break down the given polynomial:
[tex]\[ -24x^7 - 12x^2 - 9x + 6 \][/tex]
- The first term is [tex]\( -24x^7 \)[/tex] with an exponent of 7.
- The second term is [tex]\( -12x^2 \)[/tex] with an exponent of 2.
- The third term is [tex]\( -9x \)[/tex] with an exponent of 1.
- The last term is [tex]\( +6 \)[/tex] with an exponent of 0.
The exponents are listed in descending order: 7, 2, 1, and 0.
2. Examine the Coefficients:
The coefficients (numerical factors) of each term are:
- [tex]\( -24 \)[/tex] (for [tex]\( x^7 \)[/tex]),
- [tex]\( -12 \)[/tex] (for [tex]\( x^2 \)[/tex]),
- [tex]\( -9 \)[/tex] (for [tex]\( x \)[/tex]),
- [tex]\( 6 \)[/tex] (constant term).
The order of coefficients does not define the standard form of a polynomial.
3. Check if the Constant Should be First:
Another consideration might be whether the constant term should be the first term. In standard polynomial notation, the constant appears last, not first.
4. Check for Simplification:
Simplification typically means combining like terms or factorization, if possible. The given polynomial:
[tex]\[ -24x^7 - 12x^2 - 9x + 6 \][/tex]
cannot be simplified further as it has no like terms (terms that could be combined).
### Conclusion:
Based on these analyses, the polynomial [tex]\( -24x^7 - 12x^2 - 9x + 6 \)[/tex] is in standard form because its terms are in order from the highest exponent to the lowest exponent. Thus, the correct statement is:
It is in standard form because the exponents are in order from highest to lowest.
### Step-by-Step Analysis:
1. Examine the Exponents:
A polynomial is said to be in standard form if its terms are arranged in descending order of the exponents. Let's break down the given polynomial:
[tex]\[ -24x^7 - 12x^2 - 9x + 6 \][/tex]
- The first term is [tex]\( -24x^7 \)[/tex] with an exponent of 7.
- The second term is [tex]\( -12x^2 \)[/tex] with an exponent of 2.
- The third term is [tex]\( -9x \)[/tex] with an exponent of 1.
- The last term is [tex]\( +6 \)[/tex] with an exponent of 0.
The exponents are listed in descending order: 7, 2, 1, and 0.
2. Examine the Coefficients:
The coefficients (numerical factors) of each term are:
- [tex]\( -24 \)[/tex] (for [tex]\( x^7 \)[/tex]),
- [tex]\( -12 \)[/tex] (for [tex]\( x^2 \)[/tex]),
- [tex]\( -9 \)[/tex] (for [tex]\( x \)[/tex]),
- [tex]\( 6 \)[/tex] (constant term).
The order of coefficients does not define the standard form of a polynomial.
3. Check if the Constant Should be First:
Another consideration might be whether the constant term should be the first term. In standard polynomial notation, the constant appears last, not first.
4. Check for Simplification:
Simplification typically means combining like terms or factorization, if possible. The given polynomial:
[tex]\[ -24x^7 - 12x^2 - 9x + 6 \][/tex]
cannot be simplified further as it has no like terms (terms that could be combined).
### Conclusion:
Based on these analyses, the polynomial [tex]\( -24x^7 - 12x^2 - 9x + 6 \)[/tex] is in standard form because its terms are in order from the highest exponent to the lowest exponent. Thus, the correct statement is:
It is in standard form because the exponents are in order from highest to lowest.
