Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
