Answer :
To determine the proper way to write the answer for the given polynomial problem, we need to examine each of the expressions provided:
1. Expression 1: [tex]\(2x^4 - x^3 - 25x^2 - 12x - 0\)[/tex]
2. Expression 2: [tex]\(2x^3 - x^2 - 25x - 12\)[/tex]
3. Expression 3: [tex]\(2x^4 - x^3 - 25x^2 - 12x \div 0\)[/tex]
4. Expression 4: [tex]\(2x^3 - x^2 - 25x + 12\)[/tex]
Let's consider each:
- Expression 1 is a polynomial written in standard form, with terms ordered by descending powers of [tex]\(x\)[/tex]. The "-0" at the end is redundant but does not affect the polynomial.
- Expression 2 is another polynomial in standard form, with descending powers of [tex]\(x\)[/tex].
- Expression 3 includes division by zero, which is undefined in mathematics. Since dividing by zero is not possible, this expression is invalid.
- Expression 4 is yet another polynomial, also in standard form, with descending powers of [tex]\(x\)[/tex].
Given the requirement to remove expressions with undefined mathematical operations (like division by zero), we disregard Expression 3.
Thus, the valid expressions without mathematical issues are:
1. [tex]\(2x^4 - x^3 - 25x^2 - 12x - 0\)[/tex]
2. [tex]\(2x^3 - x^2 - 25x - 12\)[/tex]
3. [tex]\(2x^3 - x^2 - 25x + 12\)[/tex]
These expressions are properly written as polynomials, avoiding any undefined operations.
1. Expression 1: [tex]\(2x^4 - x^3 - 25x^2 - 12x - 0\)[/tex]
2. Expression 2: [tex]\(2x^3 - x^2 - 25x - 12\)[/tex]
3. Expression 3: [tex]\(2x^4 - x^3 - 25x^2 - 12x \div 0\)[/tex]
4. Expression 4: [tex]\(2x^3 - x^2 - 25x + 12\)[/tex]
Let's consider each:
- Expression 1 is a polynomial written in standard form, with terms ordered by descending powers of [tex]\(x\)[/tex]. The "-0" at the end is redundant but does not affect the polynomial.
- Expression 2 is another polynomial in standard form, with descending powers of [tex]\(x\)[/tex].
- Expression 3 includes division by zero, which is undefined in mathematics. Since dividing by zero is not possible, this expression is invalid.
- Expression 4 is yet another polynomial, also in standard form, with descending powers of [tex]\(x\)[/tex].
Given the requirement to remove expressions with undefined mathematical operations (like division by zero), we disregard Expression 3.
Thus, the valid expressions without mathematical issues are:
1. [tex]\(2x^4 - x^3 - 25x^2 - 12x - 0\)[/tex]
2. [tex]\(2x^3 - x^2 - 25x - 12\)[/tex]
3. [tex]\(2x^3 - x^2 - 25x + 12\)[/tex]
These expressions are properly written as polynomials, avoiding any undefined operations.
