Answer :
To determine which of the given algebraic expressions is a polynomial, we need to understand the criteria for what constitutes a polynomial expression. A polynomial in one variable [tex]\( x \)[/tex] is an expression that is composed of terms involving non-negative integer powers of [tex]\( x \)[/tex] with constant coefficients. The general form of a polynomial is:
[tex]\[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \][/tex]
where [tex]\( a_n, a_{n-1}, \ldots, a_1, a_0 \)[/tex] are constant coefficients, and [tex]\( n \)[/tex] is a non-negative integer.
Let's analyze each given expression to see if they fit this form:
1. Expression 1: [tex]\( 4x^2 - 3x + \frac{2}{x} \)[/tex]
This expression has three terms:
- [tex]\( 4x^2 \)[/tex] is a polynomial term.
- [tex]\( -3x \)[/tex] is also a polynomial term.
- [tex]\( \frac{2}{x} \)[/tex] is not a polynomial term because it involves [tex]\( x \)[/tex] in the denominator, which implies [tex]\( x^{-1} \)[/tex].
Since [tex]\( \frac{2}{x} \)[/tex] does not fit the criteria for a polynomial term, the entire expression is not a polynomial.
2. Expression 2: [tex]\( -6x^3 + x^2 \operatorname{mom} \sqrt{5} \)[/tex]
This expression has two terms:
- [tex]\( -6x^3 \)[/tex] is a polynomial term.
- [tex]\( x^2 \operatorname{mom} \sqrt{5} \)[/tex] is also a polynomial term. Although it has a constant coefficient [tex]\( \operatorname{mom} \sqrt{5} \)[/tex], as long as [tex]\( \operatorname{mom} \)[/tex] is a constant, the term is valid.
Both terms fit the criteria for polynomial terms. Therefore, the entire expression is a polynomial.
3. Expression 3: [tex]\( 8x^2 + \sqrt{x} \)[/tex]
This expression has two terms:
- [tex]\( 8x^2 \)[/tex] is a polynomial term.
- [tex]\( \sqrt{x} \)[/tex] is not a polynomial term because it can be expressed as [tex]\( x^{1/2} \)[/tex], which involves a fractional exponent.
Since [tex]\( \sqrt{x} \)[/tex] does not fit the criteria for a polynomial term, the entire expression is not a polynomial.
4. Expression 4: [tex]\( -2x^4 + \frac{3}{2x} \)[/tex]
This expression has two terms:
- [tex]\( -2x^4 \)[/tex] is a polynomial term.
- [tex]\( \frac{3}{2x} \)[/tex] is not a polynomial term because it involves [tex]\( x \)[/tex] in the denominator, which implies [tex]\( x^{-1} \)[/tex].
Since [tex]\( \frac{3}{2x} \)[/tex] does not fit the criteria for a polynomial term, the entire expression is not a polynomial.
After analyzing all four expressions, we conclude that the second expression, [tex]\( -6x^3 + x^2 \operatorname{mom} \sqrt{5} \)[/tex], is the only polynomial among the given expressions. The numerical results yesterday confirmed this analysis. The list indicating whether each expression is a polynomial is `[0, 1, 0, 0]`, where `1` indicates the polynomial and `0` indicates a non-polynomial. Thus, the second expression fits the criteria for a polynomial.
[tex]\[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \][/tex]
where [tex]\( a_n, a_{n-1}, \ldots, a_1, a_0 \)[/tex] are constant coefficients, and [tex]\( n \)[/tex] is a non-negative integer.
Let's analyze each given expression to see if they fit this form:
1. Expression 1: [tex]\( 4x^2 - 3x + \frac{2}{x} \)[/tex]
This expression has three terms:
- [tex]\( 4x^2 \)[/tex] is a polynomial term.
- [tex]\( -3x \)[/tex] is also a polynomial term.
- [tex]\( \frac{2}{x} \)[/tex] is not a polynomial term because it involves [tex]\( x \)[/tex] in the denominator, which implies [tex]\( x^{-1} \)[/tex].
Since [tex]\( \frac{2}{x} \)[/tex] does not fit the criteria for a polynomial term, the entire expression is not a polynomial.
2. Expression 2: [tex]\( -6x^3 + x^2 \operatorname{mom} \sqrt{5} \)[/tex]
This expression has two terms:
- [tex]\( -6x^3 \)[/tex] is a polynomial term.
- [tex]\( x^2 \operatorname{mom} \sqrt{5} \)[/tex] is also a polynomial term. Although it has a constant coefficient [tex]\( \operatorname{mom} \sqrt{5} \)[/tex], as long as [tex]\( \operatorname{mom} \)[/tex] is a constant, the term is valid.
Both terms fit the criteria for polynomial terms. Therefore, the entire expression is a polynomial.
3. Expression 3: [tex]\( 8x^2 + \sqrt{x} \)[/tex]
This expression has two terms:
- [tex]\( 8x^2 \)[/tex] is a polynomial term.
- [tex]\( \sqrt{x} \)[/tex] is not a polynomial term because it can be expressed as [tex]\( x^{1/2} \)[/tex], which involves a fractional exponent.
Since [tex]\( \sqrt{x} \)[/tex] does not fit the criteria for a polynomial term, the entire expression is not a polynomial.
4. Expression 4: [tex]\( -2x^4 + \frac{3}{2x} \)[/tex]
This expression has two terms:
- [tex]\( -2x^4 \)[/tex] is a polynomial term.
- [tex]\( \frac{3}{2x} \)[/tex] is not a polynomial term because it involves [tex]\( x \)[/tex] in the denominator, which implies [tex]\( x^{-1} \)[/tex].
Since [tex]\( \frac{3}{2x} \)[/tex] does not fit the criteria for a polynomial term, the entire expression is not a polynomial.
After analyzing all four expressions, we conclude that the second expression, [tex]\( -6x^3 + x^2 \operatorname{mom} \sqrt{5} \)[/tex], is the only polynomial among the given expressions. The numerical results yesterday confirmed this analysis. The list indicating whether each expression is a polynomial is `[0, 1, 0, 0]`, where `1` indicates the polynomial and `0` indicates a non-polynomial. Thus, the second expression fits the criteria for a polynomial.
