Definition of a Polynomial

The governing concept is the definition and degree of a polynomial. A polynomial in x may contain only non-negative integer powers of x. In option A, the exponents are 5, 2, and 0. All are non-negative integers, and the greatest exponent is 5, so x⁵ − 2x² + 6 is a polynomial of degree 5. Option B has exponent 5/2, which is fractional and therefore invalid. Option C has the negative exponent −5. Option D can be rewritten as x⁻⁵ + 3, so it also contains a negative exponent. Those expressions are not polynomials in x. Hence option A is the only choice satisfying both requirements: being a polynomial and having highest power 5.

A polynomial containing exactly one term is called a monomial. \(4x^7\) has only one term, so it is the correct answer. \(x^2+x\) and \(x^3-1\) each have two terms. Exam tip: Count terms separated by plus (+) or minus (−) signs.

The expression \(x^2-9\) has two terms, \(x^2\) and \(-9\), so it is a binomial. Its highest exponent is \(2\), hence its degree is 2 and it is quadratic. Therefore, it is a quadratic binomial. A cubic trinomial would need degree 3 and three terms. Exam tip: classify a polynomial by checking the number of terms and the highest exponent separately.

\(5x^{-4}+2x\) is not a polynomial because the exponent of \(x\) is \(-4\). In a polynomial, exponents of variables must be zero or positive integers. Since \(x^0=1\), option C is a polynomial; a zero exponent is allowed, but a negative exponent is not. Exam tip: An expression with a negative, fractional, or irrational exponent of a variable is not a polynomial.

\(2x^3-x^2+x-6\) has four terms: \(2x^3\), \(-x^2\), \(x\), and \(-6\). Terms of a polynomial are separated by plus or minus signs. Option A has only three terms, so it is not correct. Exam tip: Always count the constant term as a separate term.

The degree of a polynomial is the greatest exponent of the variable among its non-zero terms. Here, 6x^2 has exponent 2 and -5x has exponent 1. The constant term 0 does not affect the degree, so the answer is 2. Note that 6 is a coefficient, not the degree. Exam tip: Find the non-zero term with the highest power of the variable.

A missing power does not stop an expression from being a polynomial. In (x^4+x^2+1), all present powers are valid.

A polynomial is an algebraic expression made from terms whose variable powers are whole numbers starting at zero: \\(0,1,2,3,\\ldots\\). The coefficients may be real numbers, including integers, fractions, and terminating or non-terminating decimal values. A variable may have power zero, which represents a constant term, but negative or fractional powers are not allowed in the usual school definition of a polynomial.

Thus option B correctly says that the power must be zero or a positive integer. Option A is wrong because a negative power places the variable in a denominator. Option C is wrong because a fractional power such as \\(x^{1/2}\\) is a radical expression, and option D is too restrictive: the power need not always be 1. Therefore, B is correct.

A polynomial in \\(x\\) may have positive, zero, or negative coefficients. The sign of a coefficient does not decide whether an expression is a polynomial. What matters is that the powers of \\(x\\) are non-negative whole numbers. In \\(x^2+3x-8\\), the powers are \\(2\\), \\(1\\), and \\(0\\), so every term is permitted, and the constant term is -8.

Therefore, option A is the required expression. The other choices contain \\(x^{-2}\\), \\(\\sqrt{x}=x^{1/2}\\), or \\(1/x=x^{-1}\\). These involve a negative or fractional power and therefore are not polynomials in \\(x\\), even though each also includes the negative constant -8. A negative constant is entirely acceptable in a polynomial.

The highest exponent of x is 3 because the polynomial contains the term 2x^3; therefore, it is a cubic polynomial. It has four terms: 2x^3, 5x^2, -x, and 4. Hence, it is a cubic four-term polynomial. A quadratic polynomial has highest degree 2, so option A is not correct. Exam tip: identify the degree from the highest power of x, and count terms separated by plus or minus signs.

In \(x^5+2x^3-7\), the powers of \(x\) are \(5\), \(3\), and \(0\), respectively, because the constant term \(-7\) can be written as \(-7x^0\). In option B, the power of \(x\) is \(-3\), while option C has the fractional power \(5/3\). In option D, \(\frac{2}{x^3}=2x^{-3}\). Exam tip: a constant term always has the variable raised to power \(0\).

A polynomial with real coefficients must have real-number coefficients and only non-negative integer powers of the variable. Real coefficients can include 2.5, because every terminating decimal is a real number. The important test is whether the variable powers are 0, 1, 2, and so on, with no variable in a denominator or under a radical.

In option A, \\(2.5x^2-3x+1\\) has real coefficients and powers 2, 1, and 0, so it is a polynomial. Option B contains \\(x^{-1}\\), option C contains the fractional power \\(x^{1/2}\\), and option D has x in the denominator. These violate the required form. Therefore, option A is correct.

In a polynomial, the exponents of a variable must be non-negative integers such as \(0,1,2,\ldots\). Here, \(\frac{1}{x}=x^{-1}\), so the exponent of \(x\) is \(-1\); hence the expression is not a polynomial. The term \(x^2\) and the constant \(4\) are valid polynomial terms, and having three terms is not a restriction. Exam tip: If a variable occurs in the denominator, rewrite it using a negative exponent and check it.

A polynomial may contain several terms, constants, and numerical coefficients, provided every variable power is a non-negative integer. The power 0 is allowed because a constant can be viewed as a term involving \\(x^0\\). Therefore, the presence of a constant or addition is not a problem. What matters is the form of each power of x.

In \\(x^2+1\\), the first term has power 2 and the constant term has power 0. Both powers are valid, so this is a polynomial. The other choices contain \\(x^{-2}\\), \\(\sqrt{x}=x^{1/2}\\), or \\(1/x=x^{-1}\\); these involve a negative or fractional power of x. Thus option A gives the correct example and explanation.

The governing concept is that a polynomial may be considered in one specified variable while other symbols are treated as constants or parameters. In option A, yx³ + 2x − 5 is a polynomial in x: the powers of x are 3, 1, and 0, all non-negative integers. The symbol y acts as a coefficient-like parameter multiplying x³, so its presence does not violate the polynomial condition when x is the variable under consideration. Option B contains y/x = yx⁻¹, option C contains x⁻², and option D contains √x = x^(1/2). Each of those has an invalid power of x. Therefore option A is the only correct answer. If the question instead treated both x and y as variables jointly, the expression would still be a polynomial in two variables.

In \(x^3-2x+5\), the highest power of the variable \(x\) is 3, so its degree is 3 and it is a cubic polynomial. It has three terms: \(x^3\), \(-2x\), and \(5\); therefore, it is also a trinomial. Hence, it is a cubic trinomial. A quadratic polynomial would have highest degree 2. Exam tip: find the degree from the highest exponent, then count the terms separately.

The degree of a polynomial is the greatest exponent of the variable having a non-zero coefficient. Here, the coefficient of 0x^5 is zero, so this term does not determine the degree. Among the remaining non-zero terms, 4x^2 has the greatest exponent; therefore, the degree of p(x) is 2. Option 5 is incorrect because the coefficient of x^5 is 0. Exam tip: remove zero-coefficient terms before finding the degree.

In a polynomial, the variable has fixed non-negative integer powers. In (2^x), the variable is in the exponent, so it is not a polynomial.

On substituting \(a=0\), the term \(ax^3\) becomes 0. Thus, \(p(x)=2x-1\). The highest power of \(x\) present is 1, so the degree of the polynomial is 1. It is not 3 because the coefficient of \(x^3\) is zero. Exam tip: Ignore terms whose coefficients become zero while finding the degree.

The governing concept is the definition of a polynomial in one variable. In a polynomial, the exponent of the variable in every term must be a non-negative integer such as 0, 1, 2, or 3. The cube root of x can be rewritten as x^(1/3), so the expression becomes x^2 + x^(1/3). Although x^2 is an acceptable polynomial term, the exponent 1/3 is fractional and therefore violates the polynomial definition. Thus option A states the correct reason. Having two terms is allowed, and addition is also allowed in a polynomial. The presence of x^2 is not a problem either. The issue is specifically the fractional exponent on the variable, not the number of terms or the operation joining them.

The governing concept is the structural definition of a polynomial in x. Every term of a polynomial must contain x only with a non-negative integer exponent, and x cannot occur in a denominator. Options A, B, and C satisfy this condition: their powers of x are whole numbers such as 0, 1, 2, 3, or 4. In option D, x/(x + 1), the variable appears in the denominator. This is a rational expression rather than a polynomial, and it is undefined at x = -1. It cannot be represented as a finite sum of terms with only non-negative integer powers of x. Therefore option D is correct. The number of terms and the presence of a constant term do not disqualify the other options.

The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. The terms in the given polynomial have exponents 5, 3, 1, and 0. The highest exponent is 5, so the degree is 5. Note that 9 is a coefficient, not the degree. Exam tip: identify the term with the greatest exponent of the variable.

In \(3y^2+\sqrt{5}y-8\), the powers of \(y\) are \(2\), \(1\), and \(0\), all of which are non-negative integers. \(\sqrt{5}\) is a real number, so it is a valid coefficient. A negative constant term, \(-8\), also does not prevent an expression from being a polynomial. A zero polynomial has every coefficient equal to zero, which is not true here. Exam tip: to identify a polynomial, check the exponents of the variable; coefficients may be irrational or negative.

In the polynomial \(2x^3-5x+4\), the highest power of \(x\) is \(3\), so it is a cubic polynomial. It has three terms: \(2x^3\), \(-5x\), and \(4\); therefore, it is a trinomial. Hence, it is a cubic trinomial polynomial. A linear polynomial has highest degree 1. Exam tip: first find the highest exponent, then count the terms to classify a polynomial.

In \(8z-3\), the highest power of \(z\) is 1 and the coefficient of \(z\) is non-zero, so it is a linear polynomial. \(14\) is a constant polynomial of degree 0, while \(\frac{1}{z}+2\) is not a polynomial because it contains the negative power \(z^{-1}\). Exam tip: A linear polynomial has the form \(az+b\), where \(a\ne0\).