Definition of a Polynomial

In \(9x-4\), the highest power of the variable \(x\) is \(1\), so it is a polynomial of degree \(1\), also called a linear polynomial. \(x^2-1\) has degree \(2\), \(x^3+x\) has degree \(3\), and \(7\) is a constant polynomial with degree \(0\). Exam tip: Identify the greatest exponent of the variable to find the degree of a polynomial.

\(x^2+4\) is a polynomial because the powers of the variable \(x\) are \(2\) and \(0\), both of which are non-negative integers. It has two terms, \(x^2\) and \(4\), so it is a binomial, not a monomial. \(5x^3\) is a polynomial with only one term, so it is a monomial. In \(\sqrt{x}+2\), the exponent of \(x\) is \(\frac{1}{2}\), while \(x^2+\frac{1}{x}\) contains exponent \(-1\); therefore, neither is a polynomial. Exam tip: Variable exponents in a polynomial must be \(0,1,2,\ldots\).

The greatest exponent of the variable in a polynomial is called its degree. In the given polynomial, the highest power of x is 3 because the term 2x^3 is present. Therefore, it is a cubic polynomial. A quadratic polynomial has highest power 2, not 3. Exam tip: To identify the type of a polynomial, first find the term with the highest power of the variable.

The degree of a polynomial is determined by the term with the greatest exponent of the variable. The given polynomial has the terms \(x^5\), \(x^2\), and \(1\); among these, \(x^5\) has the highest exponent, 5. Therefore, \(x^5\) determines the degree. The exponent of \(x^2\) is only 2, while the constant term \(1\) has degree 0. Exam tip: Compare the exponents of the variable in all terms and select the greatest one.

In \(3x^2-7\), the powers of \(x\) are 2 and 0, and the highest power is 2. Hence, it is a quadratic polynomial in \(x\). A linear polynomial has highest power 1, so option C is incorrect. Exam tip: The degree of a polynomial is determined by the highest power of its variable.

The degree of a polynomial in one variable is the greatest exponent of the variable. In \(x^4-3x+2\), the highest power of \(x\) is 4, so its degree is 4. Option B has degree 3 because \(x^3\) is its highest-power term. Exam tip: To find the degree, look for the greatest power of the variable; a constant term has degree 0.

In \(5x^{-3}+2\), the exponent of \(x\) is \(-3\), which is negative. In a polynomial, the exponent of a variable must be 0 or a positive integer, so this expression is not a polynomial. \(7x^0+1\) is a polynomial because \(x^0=1\), and 0 is an allowed exponent. Exam tip: while identifying polynomials, check immediately for negative, fractional, or variable exponents.

In a polynomial, the exponent of each variable must be a non-negative integer such as \(0,1,2,\ldots\). Here, \(x^{-1}=\frac{1}{x}\), so \(5x^{-1}+2\) is not a polynomial. A constant term and a numerical coefficient are allowed in polynomials, so options B and C are incorrect. Exam tip: a negative or fractional exponent of a variable means the expression is not a polynomial.

Direct answer: Option A, x^2 + yx + 3. When y is treated as a constant, it behaves like a fixed number, just as 5 would. A polynomial in x may contain powers of x that are non-negative integers: 0, 1, 2, 3 and so on. In Option A, x^2 has power 2, yx has x-power 1, and 3 has x-power 0. Thus every x-power is allowed, and the expression is a polynomial in x. Option B contains 1/x = x^-1, a negative power, so it is not a polynomial. Option C contains sqrt(x) = x^(1/2), a fractional power, so it is not a polynomial. Option D contains x^-1, again a negative power, so it fails the definition even though y^2 is constant. Step by step: treat y as fixed; inspect every x-power; accept only whole numbers zero or greater; select A. Memory cue: polynomial powers of the chosen variable must be 0, 1, 2, 3,...

In \(x+\frac{2}{x}\), \(\frac{2}{x}=2x^{-1}\). In a polynomial, every exponent of the variable must be a non-negative integer, but \(x\) has exponent \(-1\) here. Hence, option C is not a polynomial. In option D, \(x^0=1\), so the expression becomes \(7+4=11\), a constant polynomial. Exam tip: When a variable occurs in a denominator, rewrite it with a negative exponent to check whether it is a polynomial.

The degree of a polynomial is the highest power of the variable having a non-zero coefficient. Here the terms are 4x^2, -9x, and 2, and the highest power of x is 2. Therefore, the degree of the polynomial is 2. Degree 1 would apply only to a linear polynomial; the x^2 term makes that incorrect here. Exam tip: To find degree, look for the greatest exponent of the variable, not the coefficient.

A polynomial in a variable is written using non-negative whole-number powers of that variable. The letter used as the variable may be \(x\), \(z\), or another symbol; the definition does not require the variable to be called x. In \(4z^3-z+8\), the variable is z, and the powers of z are 3, 1, and 0. All are non-negative integers, so this is a polynomial in z.

Option A and option C are polynomials in x. Option D is not a polynomial in the usual school definition because it contains \(z^{-1}\), a negative power, and it also involves z. The question asks which polynomial is given in z instead of x, so option B is the intended and correct choice. Its three terms and allowed powers satisfy the polynomial definition. The supplied answer and explanation are accurate.

In a polynomial, the exponent of a variable can only be 0 or a positive whole number. Here, x^{-1}=1/x has exponent -1, so the given expression is not a polynomial. The constant term 3 and the term 7x^2 are both allowed. Exam tip: When identifying a polynomial, first check for negative or fractional exponents, or variables in denominators.

The governing concept is the definition of a polynomial in x. Every exponent of the variable must be a non-negative integer: 0, 1, 2, 3, and so on. In option A, the powers of x are 2, 1, and 0, so x² + √3x + 5 is a polynomial. The coefficient √3 is a fixed real number, and an irrational coefficient is permitted; coefficients do not have to be rational. Option B contains √x = x^(1/2), a fractional exponent. Option C contains x⁻², a negative exponent. Option D can be rewritten as 1 + x⁻¹, so it also contains a negative power. Therefore option A alone satisfies the polynomial definition.

A polynomial whose highest power is 2 has degree 2 and is called a quadratic polynomial; for example, \(3x^2-5x+1\). A linear polynomial has highest power 1, so it is not correct. Exam tip: identify a polynomial by its highest exponent.

A polynomial whose highest exponent is 3 has degree 3, so it is called a cubic polynomial. A quadratic polynomial has highest exponent 2, so it is not correct here. Exam tip: Name a polynomial according to its highest exponent.

The terms of \(x^3+x^2+x+1\) are \(x^3\), \(x^2\), \(x\), and \(1\), so it has four terms. \(x^2+x+1\) has three terms, whereas \(x^4-1\) has only two terms. Exam tip: Count terms by separating them at plus or minus signs.

A polynomial whose every coefficient is 0, and hence whose value is 0 for every x, is called the zero polynomial. Therefore, p(x)=0 is the zero polynomial. Linear, quadratic, and cubic polynomials have degrees 1, 2, and 3 respectively, whereas the degree of the zero polynomial is not defined. Exam tip: Do not assign an ordinary degree to the zero polynomial.

The expression \(x^2-5x+6\) has three terms: \(x^2\), \(-5x\), and \(6\). Its highest power is 2, so it is a quadratic polynomial; since it has three terms, it is also a trinomial. Therefore, it is a quadratic trinomial polynomial. A linear polynomial has highest power 1, so option A is not correct. Exam tip: identify the degree from the highest power first, then count the terms to classify it as a monomial, binomial, or trinomial.

The defining condition for a polynomial in x is that every exponent of x must be a non-negative integer. In option A, the terms are 2x⁶, −4x², and 9, with exponents 6, 2, and 0 respectively. All three exponents satisfy the condition, so option A is a polynomial. The absence of x⁵, x⁴, x³, or x terms does not matter; a polynomial may have missing powers. Option B contains the negative exponent −4. Option C includes 7/x = 7x⁻¹, another negative power. Option D has the fractional exponent 2/3. Therefore only option A meets the stated definition, regardless of whether every intermediate power appears.

The degree of a polynomial is the greatest power of the variable with a non-zero coefficient. Here, the highest-power term is \(5x^3\), so the degree is \(3\). The missing \(x\)-term does not affect the degree; \(8\) is a constant term, not the degree. Exam tip: To find the degree, look for the highest exponent of \(x\) among the terms that are present.

A polynomial in t is a sum of terms in which the exponent of t is a fixed non-negative integer: 0, 1, 2, 3, and so on. In option B, the term 3/t can be rewritten as 3t⁻¹. The exponent −1 is negative, so this term violates the defining rule for a polynomial. Therefore 6 + 3/t is not a polynomial, and option B is correct. The presence of a numerical fraction alone would not be a problem, because coefficients such as 1/2 are allowed. The problem specifically occurs because the variable t is in the denominator. Options A and D use only valid whole-number powers, while option C contains t⁰ = 1 and t¹, so it is also a polynomial.

In \(\frac{5}{y}+2\), we have \(\frac{5}{y}=5y^{-1}\). The exponent of \(y\) is \(-1\), whereas every variable exponent in a polynomial must be a non-negative integer. Hence, option A is not a polynomial. Options B and C have valid non-negative integral exponents, and 7 is a constant polynomial. Exam tip: If a variable appears in the denominator, rewrite it using a negative exponent to check whether the expression is a polynomial.

The degree of a polynomial is the greatest exponent of the variable among its terms. The exponents of the terms here are 6, 4, 1, and 0 respectively. Since 6 is the greatest exponent, the degree is 6. The constant term 10 has degree 0, so it does not determine the polynomial’s degree. Exam tip: Identify the exponent of the variable in every term and select the highest one.

When every coefficient of p(x) is 0, p(x)=0 is called the zero polynomial. It has no non-zero term, so its degree is not defined. Option A is incorrect because a linear polynomial has degree 1, whereas the zero polynomial has no defined degree. Exam tip: identify the zero polynomial separately; its degree is neither 0 nor 1.