Definition of a Polynomial

Correct answer: A. x⁸ + 3x³ − 2

Explanation: A polynomial does not need to contain every power between zero and its highest power. Missing powers simply have coefficient zero, so their absence does not violate the definition. In option A, x⁸ + 3x³ − 2 has powers 8, 3 and 0; powers 7, 6, 5, 4, 2 and 1 are missing, but that is allowed. All powers that appear are non-negative integers, so the expression is a polynomial of degree 8. Option B has a negative exponent, option C has the fractional exponent 8/3, and option D contains 1/x⁸, which equals x⁻⁸. Therefore those choices are not polynomials, and option A is correct.

Correct answer: C. 4

Explanation: The degree of a polynomial is the highest exponent of its variable with a non-zero coefficient. Here, the exponents of the terms are 4, 2, 1, and 0. The highest exponent is 4, so the degree is 4. The number 7 is a coefficient, not an exponent, so it cannot be the degree. Exam tip: identify the exponents of the variable and choose the greatest one.

Correct answer: A. Linear polynomial

Explanation: In \(5x+12\), the highest power of \(x\) is 1, so its degree is 1. A polynomial of degree 1 is called a linear polynomial. A quadratic polynomial has highest power 2, so it is not correct here. Exam tip: To identify the type of a polynomial, look for the highest exponent of the variable.

Correct answer: B. Quadratic polynomial

Explanation: In \(3x^2-8x+6\), the highest power of \(x\) is 2, so its degree is 2. A polynomial of degree 2 is called a quadratic polynomial. A linear polynomial has degree 1, whereas a cubic polynomial has degree 3. Exam tip: identify the highest power of the variable to determine the type of a polynomial.

Correct answer: C. Cubic polynomial

Explanation: The degree of a polynomial is the highest power of its variable. Here, the highest power of x is 3, so it is a cubic polynomial. A quadratic polynomial has highest power 2. Exam tip: To identify the type of a polynomial, first find the highest exponent of the variable.

Correct answer: A. In \(x^{1/2}+3\), the exponent of the variable is \(1/2\), which is not a non-negative integer.

Explanation: In a polynomial, every variable exponent must be a non-negative integer such as \(0,1,2,\dots\). Since \(x^{1/2}\) has exponent \(1/2\), it is not a polynomial. Negative coefficients are allowed. Exam tip: check exponents first.

Correct answer: B. Because the variable has a fractional power

Explanation: In a polynomial in x, every exponent of x must be a non-negative integer: 0, 1, 2, and so on. The expression x^{1/2} has a fractional exponent. Although x^{1/2} can be written as √x, it still does not meet the exponent rule for a polynomial. Adding the constant 9 does not remove this invalid term.

Therefore, option B gives the correct reason. The expression has two terms and addition is allowed in polynomials, so options C and D do not identify a problem. The number 9 is also a valid constant term, so option A is not the reason. The fractional power of the variable is exactly why x^{1/2} + 9 is not a polynomial in x.

Correct answer: A. 7

Explanation: The expression 7 has no x-term, but it can be written as 7x^0. Hence, it is a constant polynomial in x with degree 0. The degrees of x, x + 7, and x^2 are 1, 1, and 2 respectively. Exam tip: recognise every non-zero constant as a multiple of x^0.

Correct answer: D. 6

Explanation: 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 6, 3, 1, and 0. The greatest exponent is 6, so the degree of the polynomial is 6. Option 3 is only the exponent in the term -5x^3, not the degree of the whole polynomial. Exam tip: First write the polynomial in simplified form, then identify the highest exponent of the variable.

Correct answer: C. 4

Explanation: In a polynomial, terms are separated by plus (+) or minus (−) signs. Here the terms are \(3x^4\), \(2x^2\), \(-x\), and \(7\). Therefore, there are 4 terms. The expression \(-x\) counts as one term, so 3 is not correct. Exam tip: Count each complete expression separated by a + or − sign as one term.

Correct answer: A. \(8x^5\)

Explanation: The terms of the polynomial are \(8x^5\), \(-3x^2\), and \(1\). Their powers of \(x\) are 5, 2, and 0 respectively. Since 5 is the greatest power, \(8x^5\) is the correct term. The term \(-3x^2\) has power 2 only, and 8 is just a coefficient, not a term. Exam tip: compare the exponent of the variable in each term to identify the highest power.

Correct answer: A. Polynomial

Explanation: In \(2x^2+\sqrt{7}x-3\), the exponents of the variable \(x\) are \(2\), \(1\), and \(0\), all of which are non-negative integers. Its coefficients \(2\), \(\sqrt{7}\), and \(-3\) are real numbers. An irrational coefficient does not stop an expression from being a polynomial, so this is a polynomial. Option B is incorrect because \(\sqrt{7}\) is only an irrational coefficient, not an exponent of the variable. Exam tip: To identify a polynomial, check that every variable exponent is \(0,1,2,\ldots\).

Correct answer: C. Not a polynomial

Explanation: As \(\sqrt{x}=x^{\frac{1}{2}}\), the expression contains \(x\) with exponent \(\frac{1}{2}\). In a polynomial, variable exponents must be non-negative integers such as \(0,1,2,\ldots\). Hence, \(4x+\sqrt{x}\) is not a polynomial. While \(4x\) alone is a linear polynomial, the term \(\sqrt{x}\) makes the complete expression non-polynomial. Exam tip: Convert a square root involving a variable into a fractional exponent before checking for a polynomial.

Correct answer: C. 0

Explanation: A constant term has no variable in it. The expression \(11x^3\) contains only a term involving \(x\), with no variable-free term; therefore, its constant term is \(0\). Here, \(11\) is the coefficient, not the constant term. Exam tip: Identify the term with no variable to find the constant term.

Correct answer: D. 0

Explanation: The terms of the polynomial are x^7, x^4, and x. There is no x^6 term, so the coefficient of x^6 is 0. The number 1 is the coefficient of x, not of x^6. Exam tip: If a required power is absent from a polynomial, its coefficient is always 0.

Correct answer: A. Yes, because the powers of x are 0 and 1, which are non-negative integers.

Explanation: Yes, (2x^0-3x) is a polynomial. Since x^0=1, the expression becomes 2-3x. The powers of x are 0 and 1, and both are non-negative integers, so it satisfies the definition of a polynomial. A subtraction sign does not make an expression non-polynomial. Exam tip: In a polynomial, the exponent of a variable must be a non-negative integer such as 0, 1, 2, ... .

Correct answer: B. Yes, because the power of x is 1

Explanation: Option B is correct. A polynomial in x is an expression in which the exponent of x in every term is a non-negative integer: 0, 1, 2, and so on. The coefficients may be integers, fractions, decimals, or other real numbers. We can write the expression as (7/3)x^1 + 4x^0. The exponents are 1 and 0, so both satisfy the definition, and the highest exponent is 1; therefore it is a linear polynomial. The fraction 7/3 is only a coefficient and does not make the expression invalid. The constant 4 is also permitted because it represents 4x^0. Finally, a binomial simply has two terms, and a binomial can certainly be a polynomial. Hence option B is the only correct choice.

Correct answer: A. Yes, because its coefficients are real and the powers of \(x\) are non-negative integers

Explanation: Yes, \(-\frac{5}{6}x^2+x-8\) is a polynomial. Its coefficients may be any real numbers, so a negative fractional coefficient such as \(-\frac{5}{6}\) is allowed. The exponents of \(x\) are \(2\), \(1\), and \(0\), all of which are non-negative integers. Option B is incorrect because a fractional coefficient does not make an expression non-polynomial. Exam tip: identify a polynomial by checking that every variable exponent is a non-negative integer.

Correct answer: B. \(x^{\frac{3}{4}}+2\)

Explanation: \(x^{\frac{3}{4}}+2\) is not a polynomial because the exponent of the variable \(x\) is \(\frac{3}{4}\), which is not a non-negative integer. In a polynomial, variable exponents can only be \(0,1,2,3,\dots\). In contrast, \(x^3+4\), \(4x^2+1\), and \(3x+4\) have exponents 3, 2, and 1 respectively, so they are polynomials. Exam tip: an expression containing a variable with a fractional exponent is not a polynomial.

Correct answer: C. 2

Explanation: The degree of a polynomial is the highest exponent of its variable with a non-zero coefficient. Here the terms are 10x^2, -6x, and 5, and the highest power of x is 2. Therefore, the degree is 2. The number 10 is a coefficient, not the degree. Exam tip: check the exponent of the variable, not its coefficient.

Correct answer: B. \(1\)

Explanation: In the polynomial \(-9x+14\), the highest power of \(x\) is \(1\), since \(-9x=-9x^1\). Therefore, its degree is \(1\). The numbers \(9\) and \(14\) are a coefficient and a constant term, not powers of the variable. Exam tip: To find the degree, look for the greatest exponent of the variable, not the coefficient.

Correct answer: C. The zero polynomial degree is not defined and a non-zero constant has degree (0)

Explanation: The zero polynomial is the polynomial whose every coefficient is zero, such as p(x) = 0. Since it has no non-zero term, there is no greatest exponent to use for its degree; its degree is therefore not defined in the usual school convention. A non-zero constant polynomial, such as 5 or -2, has no variable term but is assigned degree 0.

Option C states exactly this difference and is correct. Option A wrongly gives the zero polynomial degree 0, confusing it with a non-zero constant. Option B wrongly assigns degree 1, and option D incorrectly says that neither is a polynomial. Both expressions are polynomials; they simply have different degree conventions. Thus C follows from the definitions.

Correct answer: A. \(-4x^6\)

Explanation: The leading term of a polynomial is the term with the highest exponent of the variable. In the given polynomial, \(-4x^6\) has exponent 6, which is greater than the exponents 3 in \(2x^3\), 1 in \(-x\), and 0 in the constant term \(8\). Therefore, \(-4x^6\) is the leading term. Exam tip: compare the exponents of the terms and select the term with the greatest exponent.

Correct answer: A. 12

Explanation: The term with the greatest exponent is called the leading term. Here the greatest exponent is 4, so the leading term is 12x^4 and its coefficient is 12. The number 4 is the degree of the polynomial, not its leading coefficient. Exam tip: identify the term with the highest exponent, then take its numerical coefficient.

Correct answer: B. No, because in \(\frac{5}{x}=5x^{-1}\), the exponent of \(x\) is negative

Explanation: In a polynomial, every exponent of the variable must be a non-negative integer: \(0,1,2,\ldots\). Here, \(\frac{5}{x}=5x^{-1}\), so the exponent of \(x\) is \(-1\). Therefore, \(3x^2-\frac{5}{x}\) is not a polynomial. In option A, the exponent in \(x^2\) is valid, but the other term fails the polynomial condition. Exam tip: If a variable occurs in a denominator, rewrite it using a negative exponent and check it.