Which statement best describes the polynomial [tex]\(-24x^7 - 12x^2 - 9x + 6\)[/tex]?

A. It is in standard form because the exponents are in order from highest to lowest.
B. It is in standard form because the coefficients are in order from highest to lowest.
C. It is not in standard form because the constant should be the first term.
D. It is not in standard form because it can be further simplified.



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.