Definition of a Polynomial

For any non-zero number or variable, the zero-exponent rule is \(x^0=1\), where \(x\ne0\). Therefore, \(9x^0=9\times1=9\), and the expression becomes \(9+4x\). The number 9 is the coefficient of \(x^0\), not the value of \(x^0\). Exam tip: When you see exponent 0, first ensure that the base is non-zero, then replace that power by 1.

\(3x^2+\frac{5}{8}x-2\) is a polynomial because the exponents of \(x\) are \(2\), \(1\), and \(0\), all of which are non-negative integers. Polynomial coefficients may be fractions, so \(\frac{5}{8}\) is allowed. In options B and D, \(x\) is in the denominator, which gives a negative exponent; option C has the fractional exponent \(\frac{1}{2}\). Exam tip: while identifying a polynomial, check that every variable exponent is a non-negative integer.

Terms of a polynomial are the parts separated by addition or subtraction signs. Here the terms are \(x^4\), \(-3x^2\), \(2x\), and \(-11\). Therefore, the polynomial has 4 terms. The minus signs in \(-3x^2\) and \(-11\) belong to those terms; they do not create extra terms. Exam tip: Count the expressions separated by + or − signs.

The degree of a polynomial is the greatest exponent of the variable among terms with non-zero coefficients. Here, the highest present term is 7x^5, so the degree is 5. The absence of x^4, x^3, and x^2 does not reduce the degree; 4 is only the next lower possible exponent, not the degree. Exam tip: To find the degree, look for the largest power of x with a non-zero coefficient.

The correct answer is B. The main condition for a polynomial in x is that each exponent of x must be a whole number that is zero or positive: 0, 1, 2, 3, and so on. Coefficients may be real numbers, but the variable cannot have a negative, fractional, or otherwise non-integer exponent. Option A is the opposite of the rule: negative powers such as x^{-1} are not allowed. Option B states the correct condition. Option C is wrong because a variable need not be in a denominator; in fact, a denominator containing x often gives a negative power. Option D is wrong because a power of 1/2 is fractional and represents a square-root type expression, not a polynomial power. Learn the rule before checking degree: valid exponents are non-negative integers.

In a polynomial, powers of the variable must be integers. The power in (x^{2/5}) is (\frac{2}{5}), which is not an integer.

The degree of a polynomial is the greatest exponent of the variable among terms with non-zero coefficients. Here, \(0x^5\) is a zero term, so it is ignored. Among the remaining terms, \(4x^3\) has the highest exponent, 3. Option 5 is incorrect because the coefficient of \(x^5\) is 0. Exam tip: Remove terms with zero coefficients before finding the degree.

In \(x^{-4}+8\), the exponent of \(x\) is \(-4\), which is negative. In a polynomial, variable exponents must be zero or positive integers, so this is not a polynomial. The number \(8\) is its constant term. The other options have only non-negative integer exponents of \(x\), so they are polynomials. Exam tip: A negative, fractional, or variable exponent indicates that an expression is not a polynomial.

In the polynomial \(x^2-6x+9\), the highest power of \(x\) is \(2\), so its degree is quadratic. It has three terms: \(x^2\), \(-6x\), and \(9\), making it a trinomial. Hence, it is a quadratic trinomial polynomial. A cubic trinomial would have highest degree \(3\). Exam tip: identify the highest power first, then count the terms.

In a polynomial, the exponent of a variable must be a non-negative integer, such as \(0,1,2,\ldots\). Here, \(7/y=7y^{-1}\), so the exponent of \(y\) is \(-1\); therefore, \(7/y+2\) is not a polynomial. Having two terms alone does not make an expression a polynomial, so option B is incorrect. Exam tip: when a variable appears in the denominator, rewrite it using a negative exponent and check it.

(\frac{x+4}{x}=1+\frac{4}{x}) and (\frac{4}{x}=4x^{-1}). Due to the negative power, it is not a polynomial.

A polynomial whose highest power of the variable is 3 has degree 3, so it is called a cubic polynomial. For example, \(2x^3-x+5\) is a cubic polynomial. A quadratic polynomial has highest power 2, so it is not correct here. Exam tip: identify the name of a polynomial from its highest exponent.

In a polynomial in \(x\), the exponent of \(x\) in every term must be a non-negative integer: \(0,1,2,\ldots\). Here, \(x^{-1}=\frac{1}{x}\), so \(5x^{-1}+2\) has the variable in the denominator and is not a polynomial. A constant term and having two terms are both allowed. Exam tip: first check for negative or fractional exponents when identifying polynomials.

In p(x)=2x^2-5, the highest power of x is 2, so it is a quadratic polynomial. It has two terms, 2x^2 and -5, making it a binomial. Hence, it is a quadratic binomial. A linear binomial would have highest degree 1. Exam tip: identify a polynomial by counting its terms and then finding the highest exponent of x.

The degree of a polynomial is the highest power of its variable with a non-zero coefficient. In p(x)=9x-4, the highest power of x is 1, so it is a degree 1, or linear, polynomial. The constant term -4 does not affect the degree. Exam tip: identify the highest power of the variable to determine the type of a polynomial.

In a polynomial, the exponent of a variable must be a non-negative integer, such as \(0,1,2,\ldots\). Here, \(\frac{1}{x^3}=x^{-3}\), so \(x\) has exponent \(-3\). Since the expression contains a term with a negative exponent, it is not a polynomial. Both \(x^2\) and the constant term \(5\) are allowed in a polynomial; having three terms is also not a problem. Exam tip: when a variable occurs in the denominator, rewrite it with a negative exponent to test whether the expression is a polynomial.

The degree of a polynomial is the greatest exponent of the variable in any term with a non-zero coefficient. Here, the exponents of the terms are 4, 2, and 0, and the greatest is 4 because of the term 3x^4. Therefore, the correct answer is 4. Although 2 is the exponent in x^2, it is not the highest exponent in the polynomial. Exam tip: Treat a non-zero constant term as having degree 0, then select the greatest exponent.

In \(4z^3-2z+9\), the exponents of \(z\) are \(3\), \(1\), and \(0\). All are non-negative integers, so it is a polynomial in \(z\). Options A and C contain \(z^{-1}\), while option D contains \(z^{1/2}\); an expression with a negative or fractional exponent of the variable is not a polynomial. Exam tip: check that every variable exponent is one of \(0,1,2,\ldots\).

In \(x^6+3x^4-5\), \(x^6\) has power 6 and \(3x^4\) has power 4. The constant term \(-5\) can be written as \(-5x^0\), so its power is 0. In option B, the expression contains \(x^{-4}\), while in option D, \(\frac{3}{x^2}=3x^{-2}\). Exam tip: Any non-zero constant term has power 0 in the variable.

The correct answer is C: 2x^3-5x+√7 is a polynomial with an irrational constant term. The number √7 is irrational, but it does not contain x, so it is simply a constant coefficient; irrational constants are allowed in polynomials. The powers of x are 3 and 1, both non-negative integers, and the constant term has power 0. Option A is wrong because √x means x^{1/2}, a fractional power. Option B is wrong because x^{-2} has a negative exponent, even though √3 is an allowed constant. Option C satisfies every condition and √7 is irrational. Option D is wrong because 1/x=x^{-1}, a negative power. Thus irrationality of a coefficient is not the problem; the exponent of the variable is what matters. Memory cue: irrational constant allowed, irrational or negative power of x not allowed.

The definition of a polynomial in x permits constants and terms involving x raised only to non-negative integer powers. The expression log x is a logarithmic function, not a permitted power term such as x^0, x, or x^2. Therefore x^2 + log x is not a polynomial in x. This conclusion does not depend on the fact that the expression has two terms; two-term polynomials are possible.

Option A is correct because it contains log x. The other choices satisfy the polynomial rules: x^2 + 5x + 1 has integer powers, 7x^3 - √2 includes √2 only as a constant, and x^0 + 4x simplifies to 1 + 4x. Hence the logarithmic term, rather than addition or the number of terms, is the decisive reason.

Direct answer: Option C, the zero polynomial whose degree is not defined, is correct. Each displayed term has a zero coefficient: 0x^3 = 0, 0x^2 = 0, and 0x = 0. Their sum is therefore the zero polynomial. Degree is determined by the highest power with a non-zero coefficient. Since there is no non-zero coefficient at any power, no degree can be selected under the standard school definition. Option A is wrong because the x^3 term is actually zero and does not remain in the simplified polynomial. Option B is wrong for the same reason: the x^2 term also vanishes. Option D is wrong because the polynomial is identically 0, not the constant 1. Do not confuse the highest exponent written in an unsimplified expression with the degree after zero terms are removed. Memory cue: cancel all zero-coefficient terms first; if nothing remains, it is the zero polynomial and its degree is undefined.

In \(x^6-4x^2+9\), the powers of \(x\) are 6, 2, and 0; the constant term 9 has power 0. All are even, non-negative integers, so it is a polynomial with only even powers. In option B, \(x^5\) has an odd power. Option C has a negative exponent, and in option D, \(\sqrt{x}=x^{1/2}\) has a fractional exponent, so neither is a polynomial. Exam tip: every exponent of a variable in a polynomial must be a non-negative integer.

Polynomial terms use fixed non-negative integer powers of the variable. (\sin x) is not such a term.

An expression containing a variable in a denominator is not automatically a polynomial in its original form. However, when the question permits simplification under a stated restriction, common factors may be cancelled carefully. The restriction \\(x\ne0\\) is essential here because division by x is otherwise undefined.

For option A, factor or divide each term in the numerator by x: \\(\frac{x^3+x}{x}=\frac{x^3}{x}+\frac{x}{x}=x^2+1\\), provided \\(x\ne0\\). The result has powers 2 and 0, so it is the polynomial \\(x^2+1\\). The other options still contain negative powers or do not simplify to the required expression. Therefore option A is correct.