Definition of a Polynomial

A monomial polynomial has exactly one term, with the variable raised to a non-negative integer power. The expression 3x² has only one term, so it is a monomial. x + 2 and x - 4 are binomials, while x² + x + 1 is a trinomial. Exam tip: Count the terms to identify monomials, binomials and trinomials.

A binomial polynomial has exactly two terms. The terms in \(x^2-4\) are \(x^2\) and \(-4\), so it is a binomial. \(5x\) and \(7\) are monomials, while \(x^2+x+4\) has three terms and is a trinomial. Exam tip: count the terms separated by addition or subtraction signs.

A trinomial polynomial has exactly three terms. The terms in \(x^2+3x+2\) are \(x^2\), \(3x\), and \(2\), so it is a trinomial polynomial. In contrast, \(x^2-5\) has only two terms and is a binomial. Exam tip: count terms separated by plus or minus signs.

A fraction appearing in an expression does not automatically make it non-polynomial. We must check whether the variable itself has an allowed power. A numerical fraction can be a coefficient, and coefficients such as \\(\frac{1}{2}\\) are permitted in polynomials. The important restriction concerns powers of the variable, not every fraction seen in the expression.

Rewrite the first term as \\(\frac{x}{2}=\frac{1}{2}x^1\\). Thus the power of x is 1, a non-negative integer, while 3 is a constant term with power 0. Both terms satisfy the polynomial rule, so \\(\frac{x}{2}+3\\) is a polynomial in x. Therefore option B is correct. It is not rejected merely because its coefficient is fractional.

The governing concept is that polynomial restrictions apply to the exponents of the variable, not to whether coefficients must be integers. In (1/2)x² − 7, the exponent of x in the first term is 2, and the constant term −7 can be regarded as having x-power 0. Both 2 and 0 are non-negative integers, so the expression is a polynomial, specifically a quadratic polynomial. The coefficient 1/2 is a perfectly valid numerical coefficient and does not prevent polynomial form. Therefore option A is correct. Option B incorrectly rejects fractional coefficients. Option C is wrong because constant terms are allowed, and option D is wrong because x² is an ordinary permitted polynomial term.

Answer: option B, x^(3/2) + 1. In a polynomial in x, the exponent of x in every term must be a non-negative integer: 0, 1, 2, 3, and so on. The expression x^(3/2)+1 contains the fractional exponent 3/2, so it violates this defining condition and is not a polynomial in x. Option A has exponents 3 and 0, option C has exponents 2 and 1, and option D has exponents 1 and 0; all of these are valid. The issue is not the coefficient or the constant term. It is specifically the fractional power of the variable. A useful warning: x^(3/2) may be meaningful as a function, but it still is not a polynomial term under the school definition.

The degree of a polynomial is the highest power of its variable with a non-zero coefficient. Here, the terms are 3x^2, -5x, and 6, and the highest power of x is 2. Therefore, the degree is 2. The term -5x has power 1, while the constant term 6 has power 0. Exam tip: Simplify the polynomial first, then identify the term with the highest exponent.

The degree of a polynomial is the highest power of its variable with a non-zero coefficient. Here, 8x can be written as 8x^1, while the constant term -13 has power 0. Hence, the highest power is 1, so the correct answer is 1. Note that 8 is a coefficient, not the degree. Exam tip: For degree, look at the greatest exponent of the variable, not its coefficient.

A non-zero constant polynomial, such as 5 or -3, has no variable term with a positive power. It can be written as 5 = 5x^0, so the highest power of the variable is 0. Therefore, its degree is 0. Do not confuse it with the zero polynomial, whose degree is not defined. Exam tip: The degree of every non-zero constant polynomial is always 0.

The direct answer is C: the degree of the zero polynomial is not defined. The degree of a non-zero polynomial is the greatest exponent having a non-zero coefficient. For example, 4x^2+1 has degree 2, and 7 has degree 0. The zero polynomial is different: every coefficient is zero, so there is no greatest exponent with a non-zero coefficient. Therefore its degree is not defined in the school-level convention. Option A is wrong because degree 0 belongs to a non-zero constant polynomial such as 5, not to the zero polynomial. Option B is wrong because there is no x term of highest power 1. Option C is correct and states the standard result. Option D is wrong because no non-zero x^2 term exists. Do not confuse the number 0 as a constant with the zero polynomial. Memory cue: non-zero constant has degree 0; zero polynomial has no defined degree.

In \(2x^2+3x-1\), the only variable is \(x\), with exponents 2, 1, and 0. All are non-negative integers, so it is a polynomial in \(x\) and has no \(y\). Both \(2y^2+3\) and \(xy+1\) contain \(y\), whereas \(\frac{1}{x}+1\) is not a polynomial because \(\frac{1}{x}=x^{-1}\) has a negative exponent. Exam tip: A variable in a polynomial can have only zero or positive integer exponents.

In a polynomial, the exponent of a variable must be a non-negative integer: 0, 1, 2, 3, and so on. In \(x^2+x+1\), the powers of \(x\) are 2, 1, and 0, so it is a polynomial. Options B and D have negative exponents, while option C has \(\sqrt{x}=x^{1/2}\), a fractional exponent; hence they are not polynomials. Exam tip: reject an expression as a polynomial if a variable has a negative or fractional exponent.

Direct answer: Option A, 4x^3. A polynomial is made of terms, and the leading term is the term with the greatest exponent of the variable when the terms are written in descending order of powers. In 4x^3 - 9x^2 + x - 2, the powers of x are 3, 2, 1 and 0. The greatest power is 3, so the first and leading term is 4x^3. The number 4 is its coefficient, and 3 is its exponent. Option A is correct. Option B, -9x^2, is the second term and has power 2, which is smaller than 3. Option C, x, has power 1, so it is another later term. Option D, -2, is a constant term; its x-power is 0 and it cannot lead this polynomial. Do not confuse the leading term with the leading coefficient: the leading term is 4x^3, while the leading coefficient is 4. Exam cue: compare exponents, not the size of coefficients.

The polynomial is written in descending powers: \(5x^4+2x^2-1\). Its term with the highest power is \(5x^4\), called the leading term. Therefore, its numerical coefficient, 5, is the leading coefficient. Note that 4 is the degree of the polynomial, not its leading coefficient. Exam tip: identify the term with the greatest exponent first, then write its numerical coefficient.

In a polynomial, the exponent of a variable must be a non-negative integer such as 0, 1, 2, or 3. Here, \(\frac{3}{x}=3x^{-1}\), so the exponent of \(x\) is \(-1\). Therefore, \(2x^2+\frac{3}{x}\) is not a polynomial. Although \(2x^2\) is a polynomial term, one term with a negative exponent makes the entire expression non-polynomial. Having two terms is not a problem, since a binomial can be a polynomial. Exam tip: If a variable occurs in the denominator, rewrite it with a negative exponent and check it.

(\frac{1}{x}=x^{-1}), so it is not a polynomial. Having one term is not enough; the power must also be valid.

Since \(0x=0\), \(x^2+0x+4=x^2+4\). Therefore, the term in \(x\) with coefficient zero is omitted in simplified form. \(x^2+x+4\) is incorrect because it changes the coefficient of \(x\) from 0 to 1. Exam tip: Remove only terms whose coefficient is zero; retain all other terms.

In \(3x+7\), there is no \(x^2\) term, so the coefficient of \(x^2\) is \(0\). In contrast, the coefficients of \(x^2\) in \(x^2+5\), \(2x^2+x\), and \(4x^2-1\) are \(1\), \(2\), and \(4\), respectively. Exam tip: If a term of a particular power is absent from a polynomial, its coefficient is \(0\).

In a polynomial, every variable exponent is 0, 1, 2, 3, and so on—never negative or fractional. Negative exponents indicate reciprocal terms. Exam tip: check the exponents first when identifying a polynomial.

In a polynomial, every exponent of the variable must be a non-negative integer. In 3x^2, 2x, and 1, the powers of x are 2, 1, and 0 respectively; therefore, (3x^2+2x+1) is a polynomial. Having a constant term or three terms does not make an expression non-polynomial. Exam tip: if a variable has a negative or fractional exponent, the expression is not a polynomial.

In standard form, terms are written in descending powers. In (x^3+2x^2+x+5), powers are in the order (3), (2), (1), (0).

The term containing x is -3x. The number multiplying x is its coefficient, so the coefficient of x is -3. Here, 2 is the coefficient of x², while 5 is the constant term. Exam tip: Identify the term with the exact variable and power asked in the question.

A polynomial is an algebraic expression in which every variable has an exponent that is zero or a positive integer, for example, \(3x^2-5x+7\). Therefore, option C is correct. Expressions with negative exponents, such as \(x^{-1}\), or fractional exponents, such as \(x^{1/2}\), are not polynomials; a variable in the denominator is equivalent to a negative exponent. Exam tip: check the exponents first—they must be \(0,1,2,3,\ldots\).

In a polynomial, the exponent of each variable must be zero or a positive integer. In \(x^{\frac{2}{3}}+1\), the exponent of \(x\) is \(\frac{2}{3}\), which is fractional; therefore, it is not a polynomial. In the other options, the exponents of \(x\) are 2, 1, and 3, all of which are positive integers. Exam tip: An expression with a fractional or negative exponent of a variable is not a polynomial.

The degree of a polynomial is the highest exponent of its variable with a non-zero coefficient. Here, the term x^5 has exponent 5 and coefficient 1, so the degree is 5. The term x^3 has exponent 3, so it does not determine the degree. Exam tip: identify the greatest power of the variable among the terms.