To determine which statement best describes the polynomial [tex]\( -24x^7 - 12x^2 - 9x + 6 \)[/tex], let us analyze the structure and properties of this polynomial step-by-step.
1. Identify the terms and their exponents:
- The terms of the polynomial are:
- [tex]\(-24x^7\)[/tex] with exponent 7
- [tex]\(-12x^2\)[/tex] with exponent 2
- [tex]\(-9x\)[/tex] with exponent 1
- [tex]\(6\)[/tex] with exponent 0
2. Standard Form Definition:
- A polynomial is in standard form if its terms are arranged in descending order of their exponents, from highest to lowest.
3. Analyze the given polynomial:
- The polynomial is:
- [tex]\( -24x^7 \)[/tex] (exponent 7)
- [tex]\(-12x^2 \)[/tex] (exponent 2)
- [tex]\(-9x \)[/tex] (exponent 1)
- [tex]\(6 \)[/tex] (exponent 0)
- We can see that the exponents decrease from 7 to 2 to 1 to 0, which matches the standard form arrangement.
4. Check the given statements:
- "It is in standard form because the exponents are in order from highest to lowest."
- This is correct; we verified that the exponents decrease from 7, 2, 1 to 0.
- "It is in standard form because the coefficients are in order from highest to lowest."
- This is incorrect. Standard form concerns the order of the exponents, not the coefficients. The coefficients do not need to be in any specific order.
- "It is not in standard form because the constant should be the first term."
- This is incorrect. In standard form, the constant term, if it exists, should be the last term, not the first.
- "It is not in standard form because it can be further simplified."
- This statement is incorrect. The polynomial [tex]\( -24x^7 - 12x^2 - 9x + 6 \)[/tex] is already in its simplest form because there are no like terms to combine or further simplifications to make.
So, the correct statement is:
- "It is in standard form because the exponents are in order from highest to lowest."
Thus, the best description for the given polynomial is:
It is in standard form because the exponents are in order from highest to lowest.
