To determine which of the given algebraic expressions is a polynomial, we need to recall the definition of a polynomial. A polynomial is an expression that consists of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Polynomials cannot include division by variables or roots of variables.
Let's analyze each expression step by step:
1. Expression 1: [tex]\( 4x^2 - 3x + \frac{2}{x} \)[/tex]
- The expression [tex]\( 4x^2 \)[/tex] is a polynomial term (a term with [tex]\( x \)[/tex] raised to the power of 2).
- The expression [tex]\( -3x \)[/tex] is a polynomial term (a term with [tex]\( x \)[/tex] raised to the power of 1).
- The term [tex]\(\frac{2}{x}\)[/tex] involves division by [tex]\( x \)[/tex], which means it includes [tex]\( x \)[/tex] in the denominator. This is not allowed in polynomials.
Because [tex]\(\frac{2}{x}\)[/tex] is not a polynomial term, the entire expression is not a polynomial.
2. Expression 2: [tex]\( -6x^3 + x^2 - \sqrt{5} \)[/tex]
- The expression [tex]\( -6x^3 \)[/tex] is a polynomial term (a term with [tex]\( x \)[/tex] raised to the power of 3).
- The expression [tex]\( x^2 \)[/tex] is a polynomial term (a term with [tex]\( x \)[/tex] raised to the power of 2).
- The constant [tex]\(-\sqrt{5} \)[/tex] is a polynomial term (just a constant value, even though it involves a square root, it does not have [tex]\( x \)[/tex]).
All terms in this expression are polynomial terms. Therefore, this expression is a polynomial.
3. Expression 3: [tex]\( 8x^2 + \sqrt{x} \)[/tex]
- The expression [tex]\( 8x^2 \)[/tex] is a polynomial term (a term with [tex]\( x \)[/tex] raised to the power of 2).
- The term [tex]\(\sqrt{x} \)[/tex] involves the square root of [tex]\( x \)[/tex], which is equivalent to [tex]\( x^{1/2} \)[/tex]. This is not a non-negative integer exponent.
Because [tex]\(\sqrt{x}\)[/tex] is not a polynomial term, the entire expression is not a polynomial.
4. Expression 4: [tex]\( -2x^4 + \frac{3}{2x} \)[/tex]
- The expression [tex]\( -2x^4 \)[/tex] is a polynomial term (a term with [tex]\( x \)[/tex] raised to the power of 4).
- The term [tex]\(\frac{3}{2x} \)[/tex] involves division by [tex]\( x \)[/tex]. This is not allowed in polynomials.
Because [tex]\(\frac{3}{2x}\)[/tex] is not a polynomial term, the entire expression is not a polynomial.
In summary, out of the four expressions given, only Expression 2: [tex]\( -6x^3 + x^2 - \sqrt{5} \)[/tex] is a polynomial.
