Definition of a Polynomial

Correct answer: A. \(6x^3-5\)

Explanation: Any term multiplied by \(0\) equals \(0\). Hence, \(0x^2=0\), and removing this zero term gives \(6x^3+0x^2-5=6x^3-5\). Option B incorrectly treats the coefficient of \(x^2\) as \(1\). Exam tip: Do not write terms with a zero coefficient in the simplified form.

Correct answer: B. \(3x^2+x-6\)

Explanation: In \(3x^2+x-6\), there is no \(x^4\) term, so the coefficient of \(x^4\) is \(0\). In contrast, the coefficients of \(x^4\) in \(x^4+2\), \(5x^4-x\), and \(7x^4+1\) are \(1\), \(5\), and \(7\), respectively. Exam tip: If a term of a particular power is absent in a polynomial, its coefficient is \(0\).

Correct answer: C. Because it contains the term \(x^{-1}\), in which the variable has a negative exponent

Explanation: In a polynomial, the exponent of a variable must be a non-negative integer such as 0, 1, 2, or 3. Here, \(x^{-1}=\frac{1}{x}\) has exponent −1, so the expression is not a polynomial. The term \(5x^3\) and the constant term 4 are both allowed in a polynomial. Exam tip: if a variable occurs in the denominator, it usually represents a negative exponent and the expression is not a polynomial.

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

Explanation: Yes, (7x^4-x^2+3x-2) is a polynomial because the powers of x are 4, 2, 1, and 0, all of which are non-negative integers. The constant term -2 can be written as -2x^0, so it is also valid. Having four terms or degree 4 does not prevent an expression from being a polynomial. Exam tip: in a polynomial, variable exponents must be non-negative integers such as 0, 1, 2, and so on.

Correct answer: B. -9

Explanation: The term containing x is -9x. The number multiplying x is called its coefficient, so the coefficient of x is -9. Here, 4 is the coefficient of x², while 15 is the constant term. Exam tip: Always include the sign attached to the variable when identifying a coefficient.

Correct answer: C. 8

Explanation: The degree of a polynomial is the highest power of the variable among terms with non-zero coefficients. Here, the powers of x are 8, 6, and 2, while the constant term -5 has degree 0. The greatest power is 8, so the correct answer is 8. Option 6 is only the power in the term -3x^6, not the degree of the whole polynomial. Exam tip: Ignore zero-coefficient terms and identify the highest exponent of the variable.

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

Explanation: Direct answer: Option A, yes, because a decimal coefficient is a real number, is correct. A polynomial may have real-number coefficients, including whole numbers, fractions, and terminating decimals. The expression has terms with powers x^2, x, and x^0: 0.25x^2, −5x, and 3. These powers are non-negative integers, and no variable is in a denominator or under a root. Therefore it is a polynomial in x; its degree is 2 because the highest non-zero exponent is 2. Option B is wrong because a decimal is not forbidden; 0.25 is simply a real coefficient. Option C is wrong because a polynomial can have three terms; that makes it a trinomial, not an invalid expression. Option D is wrong because x^2 is an allowed whole-number power. Option A correctly focuses on coefficients and powers. Memory cue: decimals and fractions are allowed; reject an expression when the variable has a negative, fractional, or denominator-related power.

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

Explanation: In a polynomial, the exponent of a variable must be a non-negative integer, such as \(0,1,2,\dots\). Here, \(\frac{1}{x}=x^{-1}\), so the exponent of \(x\) is \(-1\). Hence, the expression is not a polynomial. Having two terms or a zero constant term does not stop an expression from being a polynomial. Exam tip: If a variable occurs in the denominator, rewrite it using a negative exponent and check it.

Correct answer: C. 3

Explanation: The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. Here, the term 0x^2 has no effect, while in 2x^3 the exponent of x is 3. Therefore, the degree of the polynomial is 3. The constant term 5 has degree 0, so it does not determine the degree. Exam tip: Ignore terms whose coefficients are zero when finding the degree.

Correct answer: D. Because ∛x = x¹ᐟ³

Explanation: Option D is correct because the cube-root term has a fractional exponent. Using the exponent form, ∛x = x^(1/3), so the expression is x^1 + x^(1/3). The defining condition for a polynomial is that the exponent of every variable in every term must be a non-negative integer. The exponent 1 in the first term is valid, but 1/3 in the second term is fractional and therefore invalid for a polynomial in x. Addition is entirely normal in a polynomial, and a polynomial may contain two or more terms. The presence of x itself is also not a problem. Thus the decisive issue is neither the number of terms nor the operation of addition; it is the fractional exponent created by the cube root.

Correct answer: D. 0

Explanation: The polynomial 7x^2+11 has the terms 7x^2 and 11. There is no linear term in x, so the x-term can be written as 0x. Hence, the coefficient of x is 0. Note that 7 is the coefficient of x^2, not of x. Exam tip: look for the term in which the variable has power 1; if that term is absent, its coefficient is 0.

Correct answer: B. \(2x^3-6x+4\)

Explanation: In the standard form of a polynomial, terms are arranged in descending powers of the variable. Here, the highest power is \(3\), followed by \(1\), and then the constant term \(4\). Therefore, the standard form is \(2x^3-6x+4\). Option A is not in descending order, and options C and D also do not follow the standard order. Exam tip: Write the term with the highest exponent first.

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

Explanation: Option C is correct. \(\pi\), \(\sqrt{2}\), and \(-5\) are all real numbers, so they may be coefficients of a polynomial. The exponents of \(x\) are \(2\), \(1\), and \(0\), all of which are non-negative integers. Hence, the given expression is a quadratic polynomial in \(x\). Options A and B are incorrect because irrational numbers can also be coefficients. Exam tip: when identifying a polynomial, check the exponents of the variable rather than whether the coefficients are rational.

Correct answer: D. Because \(\frac{1}{x^2}=x^{-2}\) has a negative exponent of \(x\)

Explanation: In a polynomial, the exponent of every variable must be a non-negative integer. Here, \(\frac{1}{x^2}=x^{-2}\), so the exponent of \(x\) is \(-2\). Since this exponent is negative, the expression is not a polynomial. The term \(x^3\) is valid, and a polynomial does not need to have a constant term. Exam tip: check the exponent of each variable before identifying an expression as a polynomial.

Correct answer: B. 9

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In the given polynomial, the powers are 9, 4, and 0. The greatest power is 9, so the correct answer is 9. Option 4 is only the degree of the term 3x^4, not of the whole polynomial. Exam tip: A constant term such as -2 has degree 0.

Correct answer: A. 6

Explanation: The term containing x^5 is 6x^5. The numerical factor multiplying the variable part is called the coefficient, so the coefficient of x^5 is 6. Here, 5 is the exponent of x, not its coefficient. Exam tip: Locate the required power first, then identify the number multiplying that term.

Correct answer: B. Yes, because the powers of x are 2, 1, and 0

Explanation: Yes, the expression is a polynomial in x. A polynomial may have fractional or rational coefficients; the essential requirement is that the exponents of the variable are non-negative integers. Here x²/4 has exponent 2, −2x/3 has exponent 1, and 6 is a constant with exponent 0. Therefore all terms meet the definition. Fractions, subtraction, and having three terms do not disqualify an expression.

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

Explanation: The direct answer is option B: the expression is not a polynomial because sqrt{x}=x^{1/2} has a fractional power. Start with the rule: in a polynomial in x, the exponent of x in every term must be a non-negative integer such as 0, 1, 2, or 3. Here, 3x^2 is allowed because its exponent is 2; 2sqrt{x}=2x^{1/2} is not allowed because 1/2 is not an integer; and 5 is allowed because a constant has x-power 0. Therefore one invalid term makes the whole expression non-polynomial in x. Option A is wrong: 3x^2 is a valid polynomial term. Option B is correct because it identifies the fractional exponent. Option C is wrong: constants are permitted in polynomials. Option D is wrong: having three terms does not matter; a trinomial can be a polynomial. Memory cue: check exponents, not the number of terms; polynomial powers are 0, 1, 2, 3, and so on.

Correct answer: A. \(9x^4+2x-1\)

Explanation: Since \(0x^3=0\), the term \(0x^3\) contributes nothing to the polynomial and is removed. Therefore, the simplified form is \(9x^4+2x-1\). In option C, \(9x^4\) and \(2x\) have been incorrectly combined; they are unlike terms because their powers of \(x\) are different. Exam tip: add or subtract coefficients only for like terms with the same variable powers.

Correct answer: C. 0

Explanation: The polynomial 2x^5-7x^2+4 contains an x^5 term, an x^2 term, and the constant term 4, but no x^3 term. When a term of a particular power is absent, its coefficient is 0. Therefore, the coefficient of x^3 is 0. Here, -7 is the coefficient of x^2, not x^3. Exam tip: Check the power of the variable carefully before choosing a coefficient.

Correct answer: A. Yes, because (\sqrt{11}) is a real coefficient

Explanation: In a polynomial in x, the exponents of x must be non-negative integers, while the coefficients may be any real numbers. A coefficient such as √11 is a real constant; it is not a square root of the variable x. Thus the presence of a square-root symbol alone does not make an expression non-polynomial. The number of terms also has no such restriction.

The expression is √11 x^3-5x+2. The powers of x are 3, 1, and 0, all of which are non-negative integers. Its coefficients are √11, -5, and 2, all real numbers. Hence it is a polynomial in x, and option A gives the correct reason. Options B, C, and D confuse coefficients, allowed exponents, or number of terms.

Correct answer: B. x^(2/3) + 5

Explanation: The expression x^(2/3) + 5 is not a polynomial in x because the exponent 2/3 is fractional. In a polynomial, the exponent of every variable must be a non-negative integer. The other choices have exponents 2, 1, and 3 respectively, with the constant terms understood to have exponent 0; therefore they satisfy the defining rule for polynomials.

Correct answer: A. \(5x^6\)

Explanation: The leading term of a polynomial is the term with the greatest exponent of the variable. In the given polynomial, the exponents of \(x\) are \(6, 4, 2\), and \(0\). Since \(6\) is the greatest exponent, \(5x^6\) is the leading term. Although \(-4x^4\) is also a term, its exponent is only \(4\). Exam tip: Identify the term with the highest power when the polynomial is written in descending order of powers.

Correct answer: C. Because \(\frac{3}{x^4}=3x^{-4}\) has a negative exponent of \(x\)

Explanation: Since \(\frac{3}{x^4}=3x^{-4}\), the expression contains \(x\) with exponent \(-4\). In a polynomial, the exponent of a variable must be 0 or a positive integer; negative exponents are not allowed. Having two terms or no constant term does not make an expression non-polynomial. Exam tip: If a variable is in the denominator, rewrite it with a negative exponent and check it.

Correct answer: A. (1/2)x³ + 4x − 6

Explanation: Option A is correct because a fractional coefficient is allowed in a polynomial. In (1/2)x³ + 4x − 6, the powers of x are 3, 1, and 0. Each exponent is a non-negative integer, so the expression is a polynomial, specifically a cubic polynomial. The coefficient 1/2 changes the numerical value of the term but does not affect the admissibility of its exponent. Option B contains x in the denominator and is equivalent to (1/2)x^(-1) + 4, which has a negative exponent. Option C contains the fractional exponent 1/2, and option D contains the negative exponent -2. Thus A alone satisfies the defining condition for a polynomial.