Definition of a Polynomial

In a polynomial in \(x\), the exponent of \(x\) in every term must be a non-negative integer: \(0,1,2,\dots\). A constant term has exponent \(0\). For example, \(x^{-1}\) has a negative exponent and \(x^{1/2}\) has a fractional exponent, so neither is a polynomial term. Exam tip: check all variable exponents before identifying an expression as a polynomial.

In each term of a polynomial in one variable, the exponent of the variable must be a non-negative integer such as \(0, 1, 2, \ldots\). Therefore, option A is correct. For example, \(3x^2-5x+7\) is a polynomial, whereas \(x^{-1}\) has a negative exponent and \(x^{1/2}\) has a fractional exponent, so neither is a polynomial. Exam tip: check the variable's exponents first.

The degree of a polynomial is the greatest exponent of the variable among its terms. The terms are 6x^4, -3x^2, x, and -8, and the highest exponent of x is 4. Therefore, its degree is 4. The number 2 is the degree of only the term -3x^2, not of the whole polynomial. Exam tip: identify the term with the highest power of the variable having a non-zero coefficient.

The direct answer is D, constant polynomial. A polynomial is an expression made from numbers and variables, where the powers of variables are whole numbers such as 0, 1, 2, and so on. Here the expression is only 11; it contains no variable such as x. Therefore it does not change when x changes. It can be written as 11x^0 because x^0=1, so its degree is 0. Option A, zero polynomial, is wrong because the zero polynomial is exactly 0, whereas 11 is non-zero. Option B, linear polynomial, is wrong because a linear polynomial has degree 1, usually like 3x+2. Option C, quadratic polynomial, is wrong because a quadratic polynomial has degree 2, such as x^2+1. Option D is correct because every non-zero number by itself is a constant polynomial of degree 0. Remember: no variable means constant; only 0 is the zero polynomial.

In a polynomial, powers must be zero or positive integers. (-2) is an integer but not acceptable because it is negative.

A quadratic polynomial has highest degree 2. In \(4x^2-5x+9\), the greatest exponent of \(x\) is 2, so it is quadratic. \(x^3+2x+1\) is cubic, while \(7x-1\) is linear. Also, \(\sqrt{x}+2=x^{1/2}+2\) is not a polynomial because exponents of the variable in a polynomial must be non-negative integers. Exam tip: First check that all exponents are non-negative integers, then use the highest exponent to identify the type of polynomial.

A cubic polynomial has highest exponent 3. In \(2x^3-x+4\), the highest power of \(x\) is 3, so it is a cubic polynomial. \(x^2+5\) is quadratic, while \(x^{-3}+1\) is not a polynomial because it contains a negative exponent. Exam tip: Before finding a polynomial's degree, check that all exponents are non-negative integers.

(0) is the zero polynomial but its degree is not defined. In exams, keep zero polynomial and constant polynomial separate.

In a polynomial, the powers of the variable can only be non-negative integers such as 0, 1, 2, …. Here, \(\frac{1}{x}=x^{-1}\), so the exponent of x is −1. Therefore, option B is not a polynomial. In the other options, all powers of x are non-negative integers. Exam tip: if a variable appears in a denominator, rewrite it with a negative exponent to check whether it is a polynomial.

Coefficients may be real numbers, so (\sqrt{2}) is allowed. Powers of the variable must be non-negative integers.

The governing concept is the exponent condition for polynomials. When the variable appears under a square root, √x can be written as x^(1/2). Since 1/2 is not a non-negative integer, √x + 3 is not a polynomial in x, making option B correct. A square root by itself is not automatically disqualifying: √5 and √2 are fixed real numbers because they contain no variable, so they can serve as constants in options A and C. Option D is also a polynomial because its powers of x are 3 and 0. Thus the decisive distinction is whether the variable, rather than merely a number, lies inside the radical. This prevents confusion between constant irrational coefficients and variable fractional powers.

In a polynomial, terms are separated by plus (+) or minus (−) signs. Here the terms are \(7x^5\), \(-2x^3\), \(x\), and \(-4\). Therefore, there are 4 terms. The expression \(-2x^3\) is counted as one term, including its negative sign, not as two terms. Exam tip: Count the expressions separated by + or − signs to find the number of terms.

A monomial is a polynomial with exactly one term. \(6x^4\) has only one term, so it is a monomial. \(x^2+1\) has two terms, so it is a binomial. Exam tip: Count the parts separated by + or − signs to identify the number of terms.

A binomial is a polynomial with exactly two non-zero terms. \(4x^3-9\) has the terms \(4x^3\) and \(-9\), so it is a binomial. \(7x\) is a monomial, while \(x^2+3x+2\) is a trinomial. Exam tip: count the terms separated by plus or minus signs.

A trinomial is a polynomial with three non-zero terms. The terms of \(2x^2-5x+6\) are \(2x^2\), \(-5x\), and \(6\), so it is a trinomial. \(9x^2-4\) is a binomial because it has only two terms, while \(8x^5\) is a monomial. In exams, count the terms separated by addition or subtraction signs.

For a polynomial in x, each power of x must be a non-negative integer. The coefficients can be integers, fractions, or other allowed numbers; they do not need to be whole numbers. Also, a constant term is permitted, because it can be written as a coefficient multiplied by \\(x^0\\). These rules should be checked term by term.

In \\(3x^2+\frac{2}{5}x-1\\), the powers are 2, 1, and 0. All three are non-negative integers. The fraction \\(\frac{2}{5}\\) is only the coefficient of x, so it causes no problem, and the term \\(-1\\) is a valid constant term. Hence option B correctly states that the expression is a polynomial.

The governing concept is the general definition of a polynomial in one variable. Its form may be written as a₀ + a₁x + a₂x² + ... + aₙxⁿ, where the exponents 0, 1, 2, ..., n are non-negative integers and the coefficients are permitted numbers, commonly real numbers at this level. Therefore option C states the correct general condition. Negative exponents correspond to reciprocal or denominator forms and are excluded. Fractional exponents produce radical-type expressions and are also excluded. A variable does not have to be in a denominator; in fact, placing it there usually creates a negative power. Hence options A, B, and D describe conditions incompatible with the standard polynomial definition.

In a polynomial, the exponent of each variable must be a non-negative integer, such as \(0,1,2,\ldots\). Here, \(\frac{1}{x^2}=x^{-2}\), so \(x\) has exponent \(-2\). Hence, the given expression is not a polynomial. Having no constant term is not a problem; for example, \(x^2+x\) is a polynomial. Exam tip: When a variable occurs in the denominator, rewrite it using a negative exponent and check it.

In \(y^3-2y+6\), the exponents of \(y\) are 3, 1, and 0. All are non-negative integers, so it is a polynomial in \(y\). In \(\frac{4}{y}+1\), the variable in the denominator gives \(y^{-1}\), while \(y^{-2}+5\) has a negative exponent. Also, \(\sqrt{y}+3\) contains the fractional exponent \(y^{1/2}\). Exam tip: In a polynomial, exponents of the variable must be 0, 1, 2, 3, ... .

The degree of a polynomial is the greatest exponent of the variable in any of its terms. Here, the powers of t in the terms are 2, 1, and 0 respectively, so the greatest power is 2. Therefore, the correct answer is 2. The number 10 is a constant term, so it does not determine the degree. Exam tip: To find the degree, identify the highest power of the variable.

A constant polynomial has the form \(p(x)=c\), where \(c\) is a fixed number. Hence 12, -5, and \(\frac{3}{4}\) are constant polynomials. The expression \(x+1\) contains the variable \(x\), so it is not a constant polynomial; it is a linear polynomial. Exam tip: if a variable has a power of 1 or more, the expression is not a constant polynomial.

The degree of a polynomial is the greatest exponent of a term with a non-zero coefficient. Here, the coefficient of \(x^2\) is zero, so \(0x^2\) does not contribute to the degree. The highest-degree non-zero term is \(4x^3\); therefore, the degree is \(3\). Option 2 is incorrect because the coefficient of \(x^2\) is zero. Exam tip: Ignore all terms with zero coefficients before identifying the highest exponent.

A polynomial does not need to contain every possible power of x. Missing powers simply have coefficient zero. In option A, 3x⁴ + 5 may be written as 3x⁴ + 0x³ + 0x² + 0x + 5. Its visible powers are 4 and 0, both non-negative integers, so it is a polynomial even though the x¹ term is absent. Option B contains 1/x = x⁻¹, and option C contains x⁻⁴; negative powers are not allowed in a polynomial. Option D is not the intended clear polynomial form because it is written with a variable radical and can require interpretation of √(x⁴). Thus option A is the unambiguous answer. A missing term is acceptable; an invalid exponent is not.

In a polynomial, powers of the variable must be integers. (x^{1/2}) has power (\frac{1}{2}), so it is not a polynomial.

A non-zero constant polynomial has degree 0 because it has no variable term. Therefore, 8 is a polynomial of degree 0. The degree of x is 1, and x^0+x = 1+x also has degree 1. The degree of the zero polynomial 0 is generally considered undefined. Exam tip: a non-zero expression containing only a constant is a degree 0 polynomial.