Definition of a Polynomial

The degree of a polynomial is the highest power of the variable with a non-zero coefficient. Here, \(0x^6\) has coefficient zero, so this term does not contribute to the degree. The highest-degree non-zero term is \(6x^4\); therefore, the degree of \(p(x)\) is \(4\). Option 6 is incorrect because the coefficient of \(x^6\) is zero. Exam tip: Remove terms with zero coefficients before finding the degree.

7 is a non-zero constant polynomial and can be written as 7x^0. The highest power of the variable is 0, so its degree is 0. Degree 1 is for a linear polynomial, not for a constant polynomial. Exam tip: Every non-zero constant polynomial has degree 0.

In a polynomial, every exponent of the variable must be 0 or a non-negative integer. In \(5x^{-1}+2\), the exponent of \(x\) is −1, so it is not a polynomial. \(\sqrt{2}\) may be a coefficient. Exam tip: check exponents first.

When k is treated as a constant, it can serve as a coefficient. In option A, the powers of x are 3, 1, and 0, all non-negative integers, so kx³ − 2x + 8 is a polynomial in x. The other expressions contain x in a denominator, a negative power, or a square-root power, which are not allowed in a polynomial in x.

In a polynomial, addition (+) and subtraction (−) signs separate the terms. Here the terms are \(5x^3\), \(-2x^2\), \(9x\), and \(-6\). Hence, there are 4 terms. The negative sign in \(-2x^2\) belongs to that term; it does not create an extra term. Exam tip: Count each expression separated by a + or − sign as one term.

In p(x)=x^2-4x+4, the highest power of x is 2, so its degree is 2 and it is a quadratic polynomial. It has three terms: x^2, -4x, and 4; therefore, it is also a trinomial. Hence, the correct identification is a quadratic trinomial. A linear polynomial has degree 1, so option A is not correct. Exam tip: To classify a polynomial, first find its highest degree and then count its terms.

A polynomial does not need to contain every possible power of x. It may omit the linear term, provided every power that does appear is a non-negative integer. In option A, the powers are 4, 2 and 0; all are valid polynomial powers, and there is no x¹ term. Thus x⁴ + 3x² − 5 is a polynomial with no linear term, so option A is correct. Option B has the negative exponent −4, option C has the fractional exponent 1/2, and option D has x⁻¹ because 1/x equals x⁻¹. Each of those violates the definition, regardless of whether its visible expression looks algebraic.

In \(4x^2+\frac{3}{4}x-9\), the powers of \(x\) are \(2\), \(1\), and \(0\), all of which are non-negative integers. Hence, it is a polynomial. Polynomial coefficients may be fractional or negative, so \(\frac{3}{4}\) and \(-9\) do not make it invalid. It is not the zero polynomial because all its coefficients are not zero. Exam tip: while identifying a polynomial, check the powers of the variable rather than the signs of the coefficients.

In a polynomial, variable exponents must be non-negative integers such as \(0,1,2,\ldots\). Here, \(\sqrt{x}=x^{1/2}\), so the expression is not a polynomial. Exam tip: rewrite radicals as fractional powers to check quickly.

In \(2m^5-3m+1\), the powers of \(m\) are \(5\), \(1\), and \(0\), all of which are non-negative integers. Hence, it is a polynomial in \(m\). In option A, \(1/m=m^{-1}\); option B has a negative exponent; and option D has \(\sqrt{m}=m^{1/2}\). These are not polynomials. Exam tip: powers of the variable in a polynomial must be \(0,1,2,\ldots\).

When \(a=0\), the term \(ax^4\) becomes \(0\). Thus, \(p(x)=3x^2-1\). The highest power of \(x\) in the remaining polynomial is 2, so its degree is 2. Option 4 is incorrect because the \(x^4\) term has vanished. Exam tip: Substitute the parameter value first, remove zero-coefficient terms, and then identify the highest remaining exponent.

In a polynomial, powers of the variable are zero or positive integers. This is the most important condition in the definition.

The degree of a polynomial is the greatest exponent of the variable in any of its terms. Here, x^3 has exponent 3, which is greater than the exponents in x^2, x, and the constant term 5. Therefore, the degree of the polynomial is 3. Option 2 is the degree of the term 2x^2, not of the whole polynomial. Exam tip: Compare the exponents of the variable in all terms and select the greatest one.

Having real coefficients is not enough. In (\frac{4}{x}), the power of (x) is (-1), so it is not a polynomial.

The governing concept is that the sign of a coefficient does not determine whether an expression is a polynomial. A polynomial may have positive, negative or zero coefficients, but every exponent of the variable must be a non-negative integer. In option A, x² − 11 has powers 2 and 0; the constant term is −11, and both powers are permitted. Therefore, option A is correct. Option B includes x⁻², option C includes the fractional power 1/2, and option D includes x⁻¹. Those expressions are not polynomials because of their exponents, even though each also has the same negative constant term. The negative sign is acceptable; the exponent rule is decisive.

The degree of a polynomial is the highest power of its variable with a non-zero coefficient. Here, the powers of the terms are 2, 1, and 0, and the highest is 2. Hence, it is a quadratic polynomial. A linear polynomial has degree 1, so option A is incorrect. Exam tip: identify the highest exponent of the variable to determine the type of a polynomial.

A polynomial may have any real coefficient, so \(\sqrt{3}\) is allowed. The exponents of \(p\) are 2, 1 and 0, all non-negative integers. Exam tip: check variable exponents; negative or fractional exponents make an expression non-polynomial.

If the variable appears in the denominator, its power often becomes negative. Negative powers are not valid in a polynomial.

The expression contains x^{-1}, so the exponent of x is -1. In a polynomial, every exponent of the variable must be a non-negative integer; therefore, x^2+x^{-1}+1 is not a polynomial. The term x^2 and the constant term 1 are allowed in a polynomial, so they are not the reason. Exam tip: A negative, fractional, or irrational exponent of a variable means the expression is not a polynomial.

In \(1.5x^2-4x+3\), the powers of \(x\) are \(2, 1\), and \(0\), all of which are non-negative integers. Also, \(1.5\) is a decimal real coefficient, so the expression is a polynomial. Option B has a negative exponent, while option C has \(x\) raised to \(\frac12\); neither is a polynomial. Exam tip: in a polynomial, variable exponents must be \(0,1,2,\ldots\).

A polynomial may contain some powers of the variable and omit others. Its degree is the greatest exponent having a non-zero coefficient. Therefore, the absence of an x^2 term does not prevent another term from having a larger exponent. A square root or a negative exponent would create a different issue, but neither is needed here.

In option A, the polynomial is x^4+3x-2. The x^2 term is absent, so its coefficient is 0, while the coefficient of x^4 is 1, which is non-zero. The greatest exponent is therefore 4. Option B has degree 2, option C is not a polynomial because of the negative exponent, and option D is not a polynomial because of the square-root term. Thus A is correct.

In p(x)=5x^2, there is only one term, 5x^2, so it is a monomial. The highest power of x is 2; hence its degree is 2 and it is quadratic. A linear polynomial has x to the power 1, so option B is not correct. Exam tip: use the number of terms to identify monomial/binomial and the highest exponent to identify linear, quadratic, or cubic polynomials.

In option B, \(\frac{1}{x^3}=x^{-3}\). In a polynomial, the exponent of a variable must be 0 or a positive integer; a negative exponent is not allowed. Therefore, \(4+\frac{1}{x^3}\) is not a polynomial. In contrast, in option C, \(x^0=1\), so its exponent is not negative and it is a 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 polynomial \(3x^2-7\) has two terms: \(3x^2\) and \(-7\). Its degree is the greatest exponent of the variable, which is 2 because of \(x^2\). Therefore, it has 2 terms and degree 2. The number 7 is a constant term, not the degree. Exam tip: separate terms using + or − signs, then identify the highest power of the variable.

The direct answer is C: \\(n^{4/3}+2\\) is not a polynomial in n. To test an expression, inspect every exponent of the variable. In a polynomial, each exponent must be a whole number that is zero or positive. Option A has powers 4, 2, and 0, so it is a polynomial. Option B has powers 1 and 0, so it is also a polynomial. Option C contains \\(n^{4/3}\\); the exponent \\(4/3\\) is fractional, not an integer, so it does not meet the definition. Option D contains \\(n^0\\), and exponent 0 is allowed; indeed \\(n^0=1\\) for nonzero n, so this becomes a constant expression. Thus C is correct. Option A is wrong as an answer because all its powers are permitted. Option B is wrong for the same reason. Option D is wrong because zero is specifically an allowed exponent, not a forbidden one. Do not confuse a large exponent with an invalid exponent: 4 is valid, while a fractional exponent such as \\(4/3\\) is not valid in a school-level polynomial. Exam cue: a variable may appear only with powers 0, 1, 2, 3, etc.