To determine which statement best describes the polynomial
[tex]\[
-24 x^7-12 x^2-9 x+6,
\][/tex]
we need to recall what it means for a polynomial to be in standard form.
A polynomial is in standard form if its terms are written in descending order based on the exponent of the variable [tex]\( x \)[/tex].
Let's analyze the given polynomial:
1. Check the exponents of each term:
[tex]\[
-24x^7, -12x^2, -9x, +6
\][/tex]
- The term [tex]\( -24x^7 \)[/tex] has an exponent of [tex]\( 7 \)[/tex].
- The term [tex]\( -12x^2 \)[/tex] has an exponent of [tex]\( 2 \)[/tex].
- The term [tex]\( -9x \)[/tex] has an exponent of [tex]\( 1 \)[/tex].
- The constant term [tex]\( +6 \)[/tex] has an exponent of [tex]\( 0 \)[/tex].
2. Order of the exponents:
- The exponents [tex]\( 7, 2, 1, 0 \)[/tex] are in descending order.
Since the exponents of the terms are indeed in descending order, the polynomial is correctly written in standard form.
Now, let's evaluate the given statements:
1. It is in standard form because the exponents are in order from highest to lowest: This accurately describes the polynomial since the exponents [tex]\( 7, 2, 1, 0 \)[/tex] are in descending order.
2. It is in standard form because the coefficients are in order from highest to lowest: This is incorrect. The coefficients (numbers in front of [tex]\( x \)[/tex]) don't need to be in any specific order. It is the exponents of the variable [tex]\( x \)[/tex] that must be in descending order.
3. It is not in standard form because the constant should be the first term: This is incorrect. In standard form, the term with the highest exponent should come first, not the constant.
4. It is not in standard form because it can be further simplified: This is incorrect. The polynomial is already simplified. Simplification usually involves combining like terms or factoring, neither of which applies here as there are no like terms.
The best description is:
[tex]\[
\text{It is in standard form because the exponents are in order from highest to lowest.}
\][/tex]
Thus, the correct statement is:
[tex]\[
\boxed{1}
\][/tex]
