Alright, let's identify which of the given expressions are polynomials by carefully examining each term in the expressions. A polynomial is an expression involving only non-negative integer exponents of the variable.
### Expression (a)
[tex]\[3 + 2x - 5x^2 - 4x^3\][/tex]
- The constant term [tex]\(3\)[/tex] is a polynomial term.
- The term [tex]\(2x\)[/tex] has an exponent of 1, which is a non-negative integer.
- The term [tex]\(-5x^2\)[/tex] has an exponent of 2, which is a non-negative integer.
- The term [tex]\(-4x^3\)[/tex] has an exponent of 3, which is a non-negative integer.
All terms in the expression have non-negative integer exponents. Therefore, expression (a) is a polynomial.
### Expression (b)
[tex]\[\sqrt{4} x^3 + 7x^2 - 8x + 9\][/tex]
- The term [tex]\(\sqrt{4} x^3\)[/tex] simplifies since [tex]\(\sqrt{4} = 2\)[/tex], yielding [tex]\(2x^3\)[/tex], which has an exponent of 3. This is a polynomial term.
- The term [tex]\(7x^2\)[/tex] has an exponent of 2, which is a non-negative integer.
- The term [tex]\(-8x\)[/tex] has an exponent of 1, which is a non-negative integer.
- The constant term [tex]\(9\)[/tex] is a polynomial term.
All terms in the expression have non-negative integer exponents. Therefore, expression (b) is a polynomial.
### Expression (c)
[tex]\[3x^4 + 9x^3 - 7\sqrt{x} + 8\][/tex]
- The term [tex]\(3x^4\)[/tex] has an exponent of 4, which is a non-negative integer.
- The term [tex]\(9x^3\)[/tex] has an exponent of 3, which is a non-negative integer.
- The term [tex]\(-7\sqrt{x}\)[/tex] or [tex]\(-7x^{1/2}\)[/tex] has an exponent of [tex]\(\frac{1}{2}\)[/tex], which is not an integer.
Because the exponent [tex]\(\frac{1}{2}\)[/tex] in the term [tex]\(-7\sqrt{x}\)[/tex] is not a non-negative integer, expression (c) is not a polynomial.
### Expression (d)
[tex]\[4x^3 - 10x^{-1} + 1\][/tex]
- The term [tex]\(4x^3\)[/tex] has an exponent of 3, which is a non-negative integer.
- The term [tex]\(-10x^{-1}\)[/tex] has an exponent of [tex]\(-1\)[/tex], which is negative.
Because the exponent [tex]\(-1\)[/tex] in the term [tex]\(-10x^{-1}\)[/tex] is negative, expression (d) is not a polynomial.
### Summary
- Expression (a) is a polynomial.
- Expression (b) is a polynomial.
- Expression (c) is not a polynomial because it contains the term [tex]\(\sqrt{x}\)[/tex], with a non-integer exponent of [tex]\(\frac{1}{2}\)[/tex].
- Expression (d) is not a polynomial because it contains the term [tex]\(x^{-1}\)[/tex], with a negative exponent.
