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Answer :
To determine which of the given expressions are polynomials, let's first understand what a polynomial is. A polynomial is an algebraic expression made up of terms that involve constants, variables, and non-negative integer exponents of those variables. It can include operations of addition, subtraction, and multiplication.
Now, let's examine each option:
A. [tex]\(3x^3 - 19\)[/tex]
- This expression consists of two terms: [tex]\(3x^3\)[/tex] (a variable raised to the power of 3, a non-negative integer) and [tex]\(-19\)[/tex] (a constant).
- Since it involves only non-negative integer exponents, it is a polynomial.
B. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- The term [tex]\(-x^3\)[/tex] is fine as it involves a variable with a non-negative integer exponent.
- However, [tex]\(\sqrt{-x}\)[/tex] includes a square root, which means the exponent of [tex]\(x\)[/tex] is not an integer. This disqualifies it from being a polynomial.
- Therefore, it is not a polynomial.
C. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- This expression contains terms like [tex]\(\frac{3}{5}x^4\)[/tex], [tex]\(-18x^3\)[/tex], [tex]\(x^2\)[/tex], [tex]\(-10x\)[/tex], and the constant [tex]\(3.5\)[/tex].
- All variables have non-negative integer exponents.
- Hence, it is a polynomial.
D. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- The terms involved are [tex]\(-x^3\)[/tex], [tex]\(5x^2\)[/tex], [tex]\(7x\)[/tex], and [tex]\(-1\)[/tex].
- Each term has a variable with a non-negative integer exponent or is a constant.
- So, it is a polynomial.
E. [tex]\(2x^2 + 5x - 3\)[/tex]
- This expression includes the terms [tex]\(2x^2\)[/tex], [tex]\(5x\)[/tex], and [tex]\(-3\)[/tex].
- All these terms involve variables raised to non-negative integer exponents and constants.
- Thus, it is a polynomial.
Based on this analysis, the expressions that are polynomials are: A, C, D, and E.
Now, let's examine each option:
A. [tex]\(3x^3 - 19\)[/tex]
- This expression consists of two terms: [tex]\(3x^3\)[/tex] (a variable raised to the power of 3, a non-negative integer) and [tex]\(-19\)[/tex] (a constant).
- Since it involves only non-negative integer exponents, it is a polynomial.
B. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- The term [tex]\(-x^3\)[/tex] is fine as it involves a variable with a non-negative integer exponent.
- However, [tex]\(\sqrt{-x}\)[/tex] includes a square root, which means the exponent of [tex]\(x\)[/tex] is not an integer. This disqualifies it from being a polynomial.
- Therefore, it is not a polynomial.
C. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- This expression contains terms like [tex]\(\frac{3}{5}x^4\)[/tex], [tex]\(-18x^3\)[/tex], [tex]\(x^2\)[/tex], [tex]\(-10x\)[/tex], and the constant [tex]\(3.5\)[/tex].
- All variables have non-negative integer exponents.
- Hence, it is a polynomial.
D. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- The terms involved are [tex]\(-x^3\)[/tex], [tex]\(5x^2\)[/tex], [tex]\(7x\)[/tex], and [tex]\(-1\)[/tex].
- Each term has a variable with a non-negative integer exponent or is a constant.
- So, it is a polynomial.
E. [tex]\(2x^2 + 5x - 3\)[/tex]
- This expression includes the terms [tex]\(2x^2\)[/tex], [tex]\(5x\)[/tex], and [tex]\(-3\)[/tex].
- All these terms involve variables raised to non-negative integer exponents and constants.
- Thus, it is a polynomial.
Based on this analysis, the expressions that are polynomials are: A, C, D, and E.
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