Definition of a Polynomial

In a polynomial, powers of the variable must be whole numbers including zero. In (3x^2-5x+1), the powers are (2), (1), and (0).

The direct answer is option C: \(\frac{5}{x}+1\) is not a polynomial in \(x\). A polynomial is made from constants and non-negative integral powers of the variable, such as \(x^0,x^1,x^2\), combined by addition, subtraction, and multiplication. In \(5/x\), the variable is in the denominator, so \(5/x=5x^{-1}\); the exponent is negative, which is not allowed in a polynomial. Option A, \(x^2+3x\), is a polynomial because its powers are 2 and 1. Option B, \(7x-4\), is a polynomial because it has powers 1 and 0. Option C fails because of the negative power. Option D, 9, is a constant polynomial; it can be viewed as \(9x^0\). Do not be confused by the way the options are printed: the mathematical expressions being tested are the displayed expressions themselves. Memory cue: variable in a denominator usually means “not a polynomial.”

In a polynomial, powers can be like (0,1,2,3). Fractional or negative powers do not fit the definition of a polynomial.

In \(4x^3-2x+8\), the powers of \(x\) are 3, 1, and 0 respectively. Therefore, the highest power is 3. The number 8 is a constant term, whose power is taken as 0, so it is not the highest power. Exam tip: To find the degree of a polynomial, identify the greatest power of its variable.

In \(6x+11\), the highest power of \(x\) is \(1\), so its degree is 1. A polynomial of degree 1 is called a linear polynomial. A quadratic polynomial has highest power 2, so it is not correct here. Exam tip: To identify the type of a polynomial, look for the greatest exponent of the variable.

The degree of a polynomial is the highest power of its variable with a non-zero coefficient. In \(5x^2+2x-3\), the highest power of \(x\) is 2, so it is a quadratic polynomial. A linear polynomial has degree 1, while a constant polynomial has no variable. Exam tip: To identify the type of a polynomial, first find the highest power of the variable.

The degree of a polynomial is the greatest power of its variable with a non-zero coefficient. Here, the powers of the terms are 3, 2, and 0, so the highest power is 3. Therefore, it is a cubic polynomial. A quadratic polynomial has degree 2, so option B is not correct. Exam tip: To identify the type of a polynomial, first find the highest power of the variable.

(12) has no variable, so it is a constant polynomial. A non-zero constant polynomial has degree (0).

Yes, 0 is called the zero polynomial, so it is a polynomial. A polynomial need not contain a variable; constant polynomials such as 5 and 0 are also polynomials. The degree of the zero polynomial is not defined, so option D is incorrect. Exam tip: A polynomial with constant term only 0 is called the zero polynomial.

In a polynomial, exponents of variables must be non-negative integers such as 0, 1, or 2. Here, \(x^{-1}=1/x\), so the variable occurs in the denominator and the expression is not a polynomial. Exam tip: reject expressions with negative or fractional variable exponents.

In \(x^2+4x+4\), the powers of \(x\) are \(2\), \(1\), and \(0\). All are non-negative integers, so it is a polynomial in \(x\). \(\sqrt{x}=x^{1/2}\) has a fractional exponent, while \(\frac{2}{x}=2x^{-1}\) and \(x^{-1}+5\) contain negative exponents; therefore, they are not polynomials. Exam tip: variable exponents in a polynomial can only be \(0,1,2,\ldots\).

A polynomial in x is built from terms whose powers of x are non-negative whole numbers such as 0, 1, 2, or 3. A constant term is allowed, addition is allowed, and the number 3 itself causes no problem. The expression x^(1/2)+3 fails the definition because the exponent of x is 1/2, a fractional number rather than a whole-number power.

The term x^(1/2) is the square root of x, and its exponent is explicitly 1/2. Therefore x^(1/2)+3 is not a polynomial in x, making option B correct. The issue is not the presence of the constant 3 or the plus sign; both are common in polynomials. For comparison, x^2+3 would be a polynomial because the powers 2 and 0 are non-negative integers.

In a polynomial, the exponent of a variable must be 0 or a positive integer. In the given expression, the exponent of x is -3, so it is not a polynomial. Having a constant term such as 2 or having two terms does not prevent an expression from being a polynomial. Exam tip: An expression is not a polynomial if a variable has a negative, fractional, or irrational exponent.

The degree of a polynomial is the greatest exponent of the variable among terms with non-zero coefficients. Here, the exponents of x are 4, 2, and 0, and the greatest is 4. Therefore, the correct answer is 4. The values 1, 2, and 3 are not the highest exponent in this polynomial. Exam tip: A non-zero constant term has degree 0.

In (\frac{3}{x}), (x) is in the denominator, so it is like (x^{-1}). Negative powers are not accepted in polynomials.

In a polynomial, terms are separated by the plus (+) or minus (−) signs. Here the terms are \(2x^5\), \(x\), and \(-6\). Therefore, there are 3 terms. The exponent of \(x\) being 1 does not affect the number of terms; neither the sum nor the highest exponent gives the term count. Exam tip: Count each expression separated by a + or − sign as one term.

The governing concept is identifying the power of each term in a polynomial. The variable here is x. In the term x², the exponent of x is 2, so its power is 2. In 2x, the variable is actually x¹, because an exponent of 1 is usually not written; its numerical coefficient 2 does not change the power. The terms 1 and the separate number 2 are constants and therefore have power 0. Comparing the powers gives 2, 1, and 0, with 2 being the greatest. Hence x², option A, is the correct term. A common mistake is to confuse the coefficient 2 in 2x with an exponent, but coefficient and power are different features.

A linear polynomial has degree 1. In \(3x+4\), the highest power of the variable \(x\) is 1, so it is a linear polynomial. \(x^2+1\) and \(x^3-2\) are quadratic and cubic polynomials respectively, while \(5\) is a constant polynomial. In an exam, identify the highest power of the variable to determine the degree.

In \(4x^2-1\), the highest power of \(x\) is 2. Hence, its degree is 2 and it is a quadratic polynomial. \(6x+1\) is linear because its degree is 1, while \(x^3-2\) is cubic. Exam tip: identify the type of a polynomial by checking the highest exponent of the variable.

A cubic polynomial is a polynomial of degree 3. In \(x^3-4x+2\), the highest power of the variable \(x\) is 3, so it is a cubic polynomial. Option B has degree 2, option C has degree 1, and option D is a constant polynomial. Exam tip: Identify the degree of a polynomial by looking for the highest power of its variable.

\(2x^2+\sqrt{3}x+5\) is a polynomial in \(x\) because the exponents of \(x\) are \(2\), \(1\), and \(0\), all of which are non-negative integers. \(\sqrt{3}\) is a real number, so it can be a valid coefficient of \(x\). It is not a zero polynomial because its coefficients are not all zero. Exam tip: In a polynomial, variable exponents can only be \(0,1,2,\ldots\), while coefficients may be rational or irrational.

The governing definition says that a polynomial in x can contain only non-negative integer powers of x, such as 0, 1, 2, and 3. Rewrite the radical as √x = x^(1/2). Although the term 2x has the valid integer power 1, the term x^(1/2) has a fractional power. A single fractional exponent is enough to make the complete expression non-polynomial. Therefore option C is correct. It is not linear because the radical term cannot be treated as an ordinary first-degree term, and it is not quadratic because no x² term is present. It is not constant because x appears in both terms. The coefficient 2 is allowed; the fractional exponent is the decisive issue.

The polynomial \(9x^2\) has only a term containing \(x\). A constant term has no variable in it. Since no such term is present, its value is taken as \(0\). Here, \(9\) is the coefficient of \(x^2\), not the constant term. Exam tip: Identify the term without a variable to find the constant term.

The terms of (x^4+x^2+x) are x^4, x^2, and x. There is no x^3 term, so the coefficient of x^3 is 0. Although 1 is the coefficient of x and x^2, it is not the coefficient of x^3. Exam tip: The coefficient of any missing power in a polynomial is always 0.

Yes, it is a polynomial. Since x^0 = 1, the expression becomes 1 + 2x. The powers of x are 0 and 1, both of which are non-negative integers, so the expression is a polynomial. Option B is incorrect because a variable may have power 0 in a polynomial. Exam tip: In a polynomial, the powers of variables must be non-negative integers such as 0, 1, 2, and so on.