Answer :
To determine which of the given expressions are polynomials, let's first understand what a polynomial is. A polynomial is an expression made up of variables, coefficients, and exponents, where the exponents are whole numbers (i.e., non-negative integers), and the coefficients are real numbers. Let's examine each option:
A. [tex]\(3x^3 - 19\)[/tex]
- This expression consists of a term [tex]\(3x^3\)[/tex], where the exponent is a whole number (3) and a constant term [tex]\(-19\)[/tex], making it a polynomial.
B. [tex]\(2x^2 + 5x - 3\)[/tex]
- Here, the terms are [tex]\(2x^2\)[/tex], [tex]\(5x\)[/tex], and [tex]\(-3\)[/tex]. All exponents are whole numbers (2, 1, and 0, respectively), which qualifies this expression as a polynomial.
C. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- In this expression, the exponents [tex]\((4, 3, 2, 1, 0)\)[/tex] are whole numbers. The fraction [tex]\(\frac{3}{5}\)[/tex] and decimal [tex]\(3.5\)[/tex] are acceptable coefficients in a polynomial, so this is indeed a polynomial.
D. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- The terms in this expression have exponents of whole numbers (3, 2, 1, 0), making it a polynomial.
E. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- In this expression, the term [tex]\((-x)^0.5\)[/tex] (or [tex]\(\sqrt{-x}\)[/tex]) involves a square root, which means the exponent is not a whole number. Therefore, this is not a polynomial.
Based on this analysis, the expressions that are polynomials are:
- A. [tex]\(3x^3 - 19\)[/tex]
- B. [tex]\(2x^2 + 5x - 3\)[/tex]
- C. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- D. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
E is not a polynomial due to the non-whole number exponent in the term involving the square root.
A. [tex]\(3x^3 - 19\)[/tex]
- This expression consists of a term [tex]\(3x^3\)[/tex], where the exponent is a whole number (3) and a constant term [tex]\(-19\)[/tex], making it a polynomial.
B. [tex]\(2x^2 + 5x - 3\)[/tex]
- Here, the terms are [tex]\(2x^2\)[/tex], [tex]\(5x\)[/tex], and [tex]\(-3\)[/tex]. All exponents are whole numbers (2, 1, and 0, respectively), which qualifies this expression as a polynomial.
C. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- In this expression, the exponents [tex]\((4, 3, 2, 1, 0)\)[/tex] are whole numbers. The fraction [tex]\(\frac{3}{5}\)[/tex] and decimal [tex]\(3.5\)[/tex] are acceptable coefficients in a polynomial, so this is indeed a polynomial.
D. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
- The terms in this expression have exponents of whole numbers (3, 2, 1, 0), making it a polynomial.
E. [tex]\(-x^3 + \sqrt{-x}\)[/tex]
- In this expression, the term [tex]\((-x)^0.5\)[/tex] (or [tex]\(\sqrt{-x}\)[/tex]) involves a square root, which means the exponent is not a whole number. Therefore, this is not a polynomial.
Based on this analysis, the expressions that are polynomials are:
- A. [tex]\(3x^3 - 19\)[/tex]
- B. [tex]\(2x^2 + 5x - 3\)[/tex]
- C. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]
- D. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]
E is not a polynomial due to the non-whole number exponent in the term involving the square root.
