Definition of a Polynomial

Correct answer: B. No, because \(\frac{4}{x}=4x^{-1}\) has a negative exponent of \(x\)

Explanation: In a polynomial, the exponent of a variable must be a non-negative integer, such as \(0,1,2,\dots\). Here, \(\frac{4}{x}=4x^{-1}\), so the exponent of \(x\) is \(-1\). Therefore, \(6x^3+\frac{4}{x}\) is not a polynomial. Having two terms or real coefficients alone does not make an expression a polynomial. Exam tip: If a variable appears in the denominator, rewrite it with a negative exponent and check it.

Correct answer: A. (\frac{7}{x^2})

Explanation: (\frac{7}{x^2}=7x^{-2}), so the power is negative. Having one term is not enough to be a polynomial.

Correct answer: A. \(4x^3+5\)

Explanation: Since \(0x^2=0\), the term containing \(0x^2\) makes no contribution to the polynomial and can be removed. Therefore, the simplified form is \(4x^3+5\). Option B incorrectly treats the coefficient of \(x^2\) as 1 instead of 0. Exam tip: Remove any term whose coefficient is 0 while simplifying a polynomial.

Correct answer: B. \(3x^2+2x-4\)

Explanation: In \(3x^2+2x-4\), there is no \(x^5\) term, so the coefficient of \(x^5\) is \(0\). In contrast, the coefficient of \(x^5\) is \(1\) in \(x^5+1\), and it is \(2\) and \(9\) in \(2x^5-x\) and \(9x^5+7\), respectively. Exam tip: If a term of a required power is absent, its coefficient is \(0\).

Correct answer: C. Because it contains the term \(x^{-4}\), in which the exponent of \(x\) is negative

Explanation: In a polynomial, the exponent of a variable must be a zero or a positive integer. Here, the term \(x^{-4}\) has exponent \(-4\), which is negative, so the given expression is not a polynomial. The terms \(7x^2\) and the constant \(3\) are valid polynomial terms, and having three terms is not a problem. Exam tip: Check the exponents of every variable when identifying a polynomial.

Correct answer: A. Yes, because all powers of x are non-negative integers

Explanation: Yes, (6x^4-3x^2+x-10) is a polynomial because the powers of x are 4, 2, 1, and 0. All of these are non-negative integers, as required for a polynomial. Having four terms or a constant term such as -10 does not make an expression non-polynomial; x^4 is also a valid term. Exam tip: check that no variable has a negative or fractional exponent, and that no variable appears in a denominator.

Correct answer: B. -8

Explanation: The linear term containing x is -8x. The number multiplying x is its coefficient, so the coefficient of x is -8. Here, 5 is the coefficient of x², while 12 is the constant term. Exam tip: Always include the positive or negative sign when identifying a coefficient.

Correct answer: D. 8

Explanation: The degree of a polynomial is the greatest exponent of the variable in any term with a non-zero coefficient. In the given polynomial, the exponents of x are 8, 5, 2, and 0. The greatest exponent is 8, so its degree is 8. The number 5 is only the exponent in the term 4x^5, not the degree of the polynomial. Exam tip: Check the exponents of the variable in all terms before choosing the degree.

Correct answer: A. Yes, because a decimal coefficient is also a real number

Explanation: A polynomial may have real-number coefficients, and a terminating decimal such as 0.5 is a real number. In \\(0.5x^2+3x-1\\), the coefficients are 0.5, 3, and -1, while the powers of \\(x\\) are \\(2\\), \\(1\\), and \\(0\\). All these powers are non-negative whole numbers, so the expression meets the definition of a polynomial in \\(x\\).

Therefore, option A is correct. A decimal coefficient does not disqualify an expression; it may also be written as the fraction \\(1/2\\). Having three terms is allowed, and \\(x^2\\) is a standard polynomial power. The expression would fail to be a polynomial only if it contained a negative or fractional power of \\(x\\), or a variable in the denominator.

Correct answer: B. Because after simplifying, a (\frac{1}{x}) term appears

Explanation: The direct answer is B. A polynomial in x may contain powers such as x^0, x^1, x^2, and other non-negative whole powers, but x cannot occur in a denominator. Start with the expression \\(x+1)/x\\). Divide each term in the numerator by x: \\(x/x)+(1/x)=1+1/x\\). The second term is \\(1/x=x^{-1}\\), which has a negative exponent. Therefore the expression is not a polynomial in x. Option A is wrong because addition is allowed in a polynomial; for example, x+2 is a polynomial. Option B is correct because simplification produces 1/x, or x^{-1}, which violates the polynomial rule. Option C is wrong because x+1 itself is a valid linear polynomial; its appearance in a numerator causes no problem. Option D is wrong because having two terms is allowed; many polynomials have two or more terms. Also, the expression is undefined at x=0, reinforcing the denominator issue, though the essential test is the negative power. Memory cue: variable in the denominator means check carefully; it usually means not a polynomial.

Correct answer: C. 3

Explanation: The degree of a polynomial is the highest exponent of a term with a non-zero coefficient. Here, 0x² and 0x are zero terms, so they do not affect the degree. The term 2x³ has exponent 3; therefore, the degree is 3. The number 9 is a constant term and has degree 0. Exam tip: Ignore terms with zero coefficients before identifying the highest power.

Correct answer: C. 3x⁴ + x/5 − 6

Explanation: The governing definition is that a polynomial in x is a finite sum of terms whose powers of x are non-negative integers; coefficients may be real numbers, including fractions. In option C, the powers are 4, 1 and 0, all of which satisfy the definition, and 1/5 is an allowed coefficient. Option A contains x⁻², option B contains the fractional power −1/2, and option D contains x³ᐟ² because √(x³) equals x³ᐟ² in the usual formal interpretation. Negative or fractional exponents disqualify those expressions from being polynomials. Therefore, option C is the unique correct answer.

Correct answer: D. 0

Explanation: The terms of (4x^2-7) are 4x^2 and -7. There is no linear term containing x, so the coefficient of x is 0. Here, 4 is the coefficient of x^2, not of x. Exam tip: Look for the term with exactly the power of the variable asked in the question.

Correct answer: C. −5/2

Explanation: A constant polynomial has no variable term, so its value does not depend on x. A non-zero constant polynomial must also have a value other than zero. Option C, −5/2, contains no x and is not equal to zero, so it is a non-zero constant polynomial. Option A is the zero polynomial, not a non-zero constant, even though it is constant in a broad sense. Option B contains x and is therefore a linear polynomial. Option D contains x² and is a quadratic polynomial. The classification depends on both conditions: absence of the variable and a non-zero constant value. Hence option C is the only answer satisfying the complete requirement.

Correct answer: B. \(5x^4-2x+9\)

Explanation: In the standard form of a polynomial, terms are arranged in descending powers of the variable. Here, the highest power is \(4\), followed by \(1\), and then the constant term \(9\). Therefore, the correct standard form is \(5x^4-2x+9\). In option C, the \(x\)-term comes after the constant term, so the powers are not in descending order. Exam tip: place the term with the highest exponent first when writing a polynomial in standard form.

Correct answer: C. It is a polynomial because its coefficients are real and the powers of \(x\) are non-negative integers

Explanation: \(\sqrt{3}\), \(-\pi\), and \(2\) are all real numbers, so they are valid coefficients of a polynomial. The powers of \(x\) are \(3\), \(1\), and \(0\), all of which are non-negative integers. Therefore, the given expression is a polynomial. The fact that \(\pi\) and \(\sqrt{3}\) are irrational does not make the expression non-polynomial; it is the exponents of the variable that must be non-negative integers. Exam tip: check the exponents of the variable first.

Correct answer: B. Because \(\frac{1}{x^3}=x^{-3}\) gives \(x\) a negative exponent

Explanation: Since \(\frac{1}{x^3}\) can be written as \(x^{-3}\), the expression contains \(x\) with exponent \(-3\). In a polynomial, the exponent of a variable must be a non-negative integer: \(0,1,2,\ldots\); negative exponents are not allowed. The absence of a constant term does not make an expression non-polynomial, so option D is not a valid reason. Exam tip: If a variable occurs in a denominator, rewrite it using a negative exponent to check whether the expression is a polynomial.

Correct answer: B. 10

Explanation: The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. Here, the exponents of the terms are 10, 5, and 0, so the greatest exponent is 10. Therefore, the correct answer is 10. The number 5 is only the exponent in the term 2x^5, not the degree of the whole polynomial. Exam tip: Treat a non-zero constant term as having exponent 0 and select the greatest exponent.

Correct answer: B. 2

Explanation: The given polynomial is (2x^6-4x^2+8). Its term containing (x^6) is (2x^6), and the number multiplying (x^6) is 2. Hence, the coefficient of (x^6) is 2.

(-4) is the coefficient of (x^2), while 8 is the constant term. Exam tip: To find a coefficient, identify the number multiplying the required variable term.

Correct answer: C. No, because (x^{-1}) has a negative power

Explanation: For an expression to be a polynomial in \\(x\\), every exponent of \\(x\\) must be a non-negative whole number. In \\(x^2+x^{-1}+x^0\\), the terms \\(x^2\\) and \\(x^0\\) are allowed, but \\(x^{-1}\\) has a negative exponent. Indeed, \\(x^{-1}=1/x\\), which places the variable in a denominator.

One invalid term is enough to make the complete expression non-polynomial. Therefore, option C is correct. The number of terms does not decide whether an expression is a polynomial, and the presence of one valid term such as \\(x^2\\) cannot repair the invalid negative power. The correct test is to inspect every variable exponent and ensure all are non-negative integers.

Correct answer: B. Yes, because the variable powers are (2), (1), and (0)

Explanation: Fractions in the coefficients are allowed in a polynomial. The definition does not require every coefficient to be an integer. Instead, it requires the powers of the variable to be non-negative integers, and it permits addition or subtraction between valid terms. Thus the signs and the number of terms do not determine whether an expression is a polynomial.

Write the expression as \\(\frac{1}{5}x^2-\frac{3}{2}x+1\\). The powers of x are 2, 1, and 0. Each is a non-negative integer, while \\(\frac{1}{5}\\) and \\(-\frac{3}{2}\\) are simply coefficients. Therefore the expression is a polynomial in x, and option B is correct. Neither subtraction nor having three terms makes it invalid.

Correct answer: B. Because (\sqrt{x}=x^{\frac{1}{2}})

Explanation: The direct answer is B. A polynomial in x may contain powers such as 0, 1, 2 and other non-negative integers. Rewrite the radical: √x=x^{1/2}. The exponent 1/2 is fractional, not an integer, so the term 3√x does not satisfy the polynomial definition. The other term 2x^2 is allowed because its exponent is 2, and the constant 1 is allowed because it has exponent 0. Option A is wrong: x^2 is a valid polynomial term. Option B is correct because it identifies the fractional power. Option C is wrong: constants are permitted in polynomials. Option D is wrong: having three terms does not disqualify an expression; a trinomial can be a polynomial. Memory cue: polynomial powers of the variable must be whole numbers 0, 1, 2, ...; roots of x usually create fractional powers.

Correct answer: A. x² + 4x + 4, because the powers are 2, 1 and 0

Explanation: By definition, a polynomial in x is a finite expression in which the exponent of x in every term is a non-negative integer. Option A, x² + 4x + 4, has powers 2, 1 and 0, so it satisfies the definition and is a quadratic polynomial. Option B is not a polynomial because −2 is a negative exponent. Option C is not a polynomial because √x represents x¹ᐟ², a fractional exponent. Option D is not a polynomial because 1/x represents x⁻¹; the variable effectively occurs in the denominator. The reasons stated in the distractors do not establish polynomial status. Therefore, option A is the unique correct example.

Correct answer: B. Every polynomial is an algebraic expression

Explanation: The governing relationship is set inclusion: a polynomial is a special kind of algebraic expression whose variable exponents are non-negative integers and whose coefficients are appropriate numbers. Therefore every polynomial is an algebraic expression, making option B correct. The reverse is false because algebraic expressions may contain denominators with variables, radicals, or other forms not allowed in polynomials.

Correct answer: A. Variable powers are zero or positive integers and coefficients may be real.

Explanation: The governing definition is that a polynomial in a variable is a finite sum of terms whose variable exponents are non-negative integers: 0, 1, 2, and so on. Its coefficients may be real numbers, including integers, fractions, or irrational constants. Therefore option A gives the correct general condition. Option B is wrong because negative powers such as x⁻² represent a variable in the denominator. Option C is wrong because a variable under a root generally produces a fractional or otherwise non-permitted exponent, as in √x = x¹ᐟ². Option D is also wrong because a variable in the denominator gives a negative exponent. A polynomial may contain a constant term corresponding to exponent zero, but none of the other statements is required.