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Answer :
To determine which of the given expressions are polynomials, let's understand what a polynomial is. A polynomial is a mathematical expression that consists of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. With this in mind, let's check each option:
A. [tex]\(3x^3 - 19\)[/tex]: This is a polynomial. It consists of a variable [tex]\(x\)[/tex] raised to the power of 3, which is a non-negative integer, and a constant term. This fits the definition of a polynomial.
B. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]: This is also a polynomial. Each term involves [tex]\(x\)[/tex] raised to a non-negative integer exponent. The coefficients, including the fraction [tex]\(\frac{3}{5}\)[/tex] and the decimal 3.5, do not affect the polynomial nature of the expression.
C. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]: This is a polynomial. Like the previous examples, it involves [tex]\(x\)[/tex] raised to non-negative integer powers, and all terms are valid for a polynomial expression.
D. [tex]\(-x^3 + \sqrt{-x}\)[/tex]: This is not a polynomial because [tex]\(\sqrt{-x}\)[/tex] can be rewritten as [tex]\(-x^{1/2}\)[/tex], which has a fraction as an exponent. A polynomial must have terms where the exponents are non-negative integers, and this term does not meet that requirement.
E. [tex]\(2x^2 + 5x - 3\)[/tex]: This is a polynomial. It consists of terms with [tex]\(x\)[/tex] raised to the non-negative integer powers of 2 and 1, and a constant term.
In summary, options A, B, C, and E are polynomials, while option D is not because it includes a term with a fractional exponent.
A. [tex]\(3x^3 - 19\)[/tex]: This is a polynomial. It consists of a variable [tex]\(x\)[/tex] raised to the power of 3, which is a non-negative integer, and a constant term. This fits the definition of a polynomial.
B. [tex]\(\frac{3}{5}x^4 - 18x^3 + x^2 - 10x + 3.5\)[/tex]: This is also a polynomial. Each term involves [tex]\(x\)[/tex] raised to a non-negative integer exponent. The coefficients, including the fraction [tex]\(\frac{3}{5}\)[/tex] and the decimal 3.5, do not affect the polynomial nature of the expression.
C. [tex]\(-x^3 + 5x^2 + 7x - 1\)[/tex]: This is a polynomial. Like the previous examples, it involves [tex]\(x\)[/tex] raised to non-negative integer powers, and all terms are valid for a polynomial expression.
D. [tex]\(-x^3 + \sqrt{-x}\)[/tex]: This is not a polynomial because [tex]\(\sqrt{-x}\)[/tex] can be rewritten as [tex]\(-x^{1/2}\)[/tex], which has a fraction as an exponent. A polynomial must have terms where the exponents are non-negative integers, and this term does not meet that requirement.
E. [tex]\(2x^2 + 5x - 3\)[/tex]: This is a polynomial. It consists of terms with [tex]\(x\)[/tex] raised to the non-negative integer powers of 2 and 1, and a constant term.
In summary, options A, B, C, and E are polynomials, while option D is not because it includes a term with a fractional exponent.
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