Definition of a Polynomial

Correct answer: C. 4

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 4, 1, and 0, so the greatest exponent is 4. Therefore, the correct answer is 4. Note that 9 is a coefficient, not an exponent. Exam tip: To find degree, look for the highest power of the variable, not the coefficient.

Correct answer: A. Linear polynomial

Explanation: In \(3x-10\), 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 \(2\), whereas in a zero polynomial all coefficients are zero. Exam tip: To identify the type of a polynomial, look for the highest power of the variable.

Correct answer: B. Quadratic polynomial

Explanation: In \(x^2-8x+12\), 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: To identify the type of a polynomial, look for the highest power of the variable.

Correct answer: C. Cubic polynomial

Explanation: The degree of a polynomial is the highest power of its variable with a non-zero coefficient. In \(4x^3+x-6\), 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: D. Constant polynomial

Explanation: The expression (-15) has no variable and its value is a fixed non-zero number. Hence, it is a constant polynomial, with degree 0. A zero polynomial is only 0, so (-15) is not a zero polynomial. Exam tip: A polynomial containing no variable is identified as a constant polynomial.

Correct answer: A. Its degree is not defined

Explanation: The zero polynomial has all coefficients equal to 0, for example, 0. The degree of a polynomial is the greatest exponent of a non-zero term. Since the zero polynomial has no non-zero term, its degree is not defined. Degree 0 belongs to a non-zero constant polynomial such as 5. Exam tip: Do not confuse the zero polynomial with a non-zero constant polynomial.

Correct answer: B. Because the exponent of the variable is fractional

Explanation: In a polynomial, every variable exponent must be a non-negative integer. Here, the exponent of \(x\) is \(\frac{5}{2}\), which is fractional, so \(x^{\frac{5}{2}}-1\) is not a polynomial. A subtraction sign, a negative constant term, or two terms does not make an expression non-polynomial. Exam tip: check that each variable exponent is 0 or a positive integer.

Correct answer: A. Because the variable x has a negative exponent

Explanation: In a polynomial, every exponent of a variable must be a non-negative integer. Here, the exponent of x is -2, so the term x^{-2} = 1/x^2 cannot be a polynomial term. Although 7x is a valid polynomial term, one invalid exponent makes the whole expression non-polynomial. Exam tip: Check for negative or fractional exponents before identifying an expression as a polynomial.

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, 2, and 0; the greatest is 6. Therefore, the correct answer is 6. The number 2 is only the exponent in the term -4x^2, not the degree of the whole polynomial. Exam tip: treat a non-zero constant term as having exponent 0, then identify the greatest exponent.

Correct answer: C. 4

Explanation: In a polynomial, the parts separated by addition or subtraction signs are called terms. Here, the terms are \(7x^3\), \(2x^2\), \(-x\), and \(5\). Therefore, there are 4 terms. Remember that \(-x\) is one negative term, not something to omit because of the minus sign. Exam tip: Count every positive or negative algebraic part as one complete term.

Correct answer: A. \(2x^3\)

Explanation: The terms of the polynomial are \(2x^3\), \(x^2\), and \(4\). Their powers of \(x\) are 3, 2, and 0 respectively. Therefore, \(2x^3\) has the highest power. The term \(x^2\) has power 2 only. Exam tip: Compare the exponents of the variable to identify the term with the highest power.

Correct answer: A. Polynomial

Explanation: This is a polynomial because the powers of \(x\) are \(2\), \(1\), and \(0\), all of which are non-negative integers. Since \(\sqrt{5}\) is a real number, it can be a valid coefficient of \(x\). A zero polynomial has all coefficients equal to zero, which is not the case here. Exam tip: in a polynomial, variable powers must be \(0,1,2,\ldots\); coefficients may also be irrational.

Correct answer: C. Not a polynomial

Explanation: A polynomial in \(x\) can contain only non-negative integer powers of \(x\). Its terms may have coefficients and may be combined by addition or subtraction, but fractional powers such as \(x^{1/2}\) are not allowed in a polynomial. Since \(\sqrt{x}=x^{1/2}\), the first term in \(\sqrt{x}+5x^2\) has a fractional exponent.

Although \(5x^2\) by itself is a quadratic polynomial, adding \(\sqrt{x}\) makes the complete expression fail the polynomial definition. It is therefore not a polynomial, and it cannot be classified as linear or quadratic. It is also not constant because it changes with x. Hence option C is correct. The supplied explanation identifies the fractional power precisely and gives the correct conclusion.

Correct answer: C. 0

Explanation: A constant term contains no variable. The only term in \(12x^3\) contains \(x\), so there is no separate constant term. Hence, its constant term is \(0\). Here, \(12\) is the coefficient, not the constant term. Exam tip: A term with the variable raised to the power \(0\) is the constant term.

Correct answer: D. 0

Explanation: The polynomial x^5+x^3+x contains the terms x^5, x^3, and x, but it has no x^4 term. When a term of a particular degree is absent, its coefficient is taken as 0. Therefore, the coefficient of x^4 is 0. Choosing 1 would be incorrect because 1 is the coefficient of x, not x^4. Exam tip: Look for the exact power named in the question before identifying its coefficient.

Correct answer: A. Yes, because \(x^0=1\) and the exponent of \(x\) in \(1-4x\) is a non-negative integer

Explanation: Since \(x^0=1\), the given expression becomes \(1-4x\). The exponents of \(x\) are \(0\) and \(1\), both of which are non-negative integers; hence it is a polynomial. An exponent of zero is allowed, and neither subtraction nor having two terms prevents an expression from being a polynomial. Exam tip: check that every variable exponent is \(0,1,2,\ldots\).

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

Explanation: In (\frac{3x}{5}), the power of (x) is (1) and the coefficient is (\frac{3}{5}). A fractional coefficient is valid in a polynomial.

Correct answer: A. Yes, because its coefficients are real numbers and the exponent of \(x\) is a non-negative integer.

Explanation: Yes, \(-\frac{2}{3}x^3+4\) is a polynomial. The coefficients of a polynomial may be any real numbers, including negative numbers and fractions. Also, the exponent of the variable must be a non-negative integer; here, the exponent of \(x\) is \(3\). A fractional coefficient is allowed, whereas a fractional exponent such as \(x^{1/2}\) is not allowed in a polynomial. Exam tip: check the exponents, not the form of the coefficients—exponents must be \(0,1,2,\ldots\).

Correct answer: B. \(x^{\frac{4}{5}}+2\)

Explanation: The correct answer is \(x^{\frac{4}{5}}+2\) because the power of \(x\) is \(\frac{4}{5}\), a fractional exponent. In a polynomial, every exponent of the variable must be a non-negative integer such as \(0,1,2,\ldots\). The exponents in \(x^4+5\), \(5x^2+1\), and \(4x+5\) are 4, 2, and 1 respectively, so they are polynomials. Exam tip: To identify a polynomial, first check the exponents of its variables.

Correct answer: C. 2

Explanation: The degree of a polynomial is the highest exponent of its variable. Here, the exponents in the terms are 2, 1, and 0, and the greatest is 2. Therefore, the degree of the polynomial is 2. The number 6 is a coefficient, not the degree. Exam tip: identify the greatest power of the variable.

Correct answer: B. 1

Explanation: The degree of a polynomial is the greatest power of its variable with a non-zero coefficient. Here, 13x can be written as 13x^1, while the constant term 2 has power 0. Therefore, the highest power is 1, so the correct answer is 1. The number 13 is only the coefficient of x, not the degree. Exam tip: For degree, look at the highest exponent of the variable, not its coefficient.

Correct answer: A. 0

Explanation: 23 is a non-zero constant polynomial. It can be written as 23x^0, so the highest power of x is 0 and its degree is 0. Degree 1 belongs to linear polynomials such as 2x + 3. Exam tip: only the zero polynomial has an undefined degree; every non-zero constant has degree 0.

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

Explanation: The degree of a polynomial is the greatest exponent of the variable having a non-zero coefficient. For a non-zero constant polynomial such as 5 or -3, there is no visible variable term, but it can be viewed as a term with exponent \\(0\\), since \\(x^0=1\\). Therefore, every non-zero constant polynomial has degree 0.

The zero polynomial is different because all its coefficients are zero. It has no highest exponent with a non-zero coefficient, so its degree is not defined in the usual school definition. Consequently, option C gives the correct difference. Option A incorrectly assigns degree 0 to the zero polynomial, while option D is wrong because the zero expression is still accepted as a polynomial.

Correct answer: A. \(x^5+x^2+1\)

Explanation: In a polynomial, the powers of a variable must be 0 or positive integers. In \(x^5+x^2+1\), the powers of \(x\) are 5, 2, and 0; the constant term \(1\) has power 0. Hence, option A is correct. Option B has a negative power, option C has a fractional power, and in option D, \(\frac{1}{x}=x^{-1}\), which has a negative power. Exam tip: To identify a polynomial, check that no variable has a negative or fractional exponent.

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

Explanation: The leading term of a polynomial is the term with the highest power of the variable. The given polynomial has terms involving \(x^4\), \(x^2\), \(x\), and the constant \(1\). Since the highest power is 4, \(-3x^4\) is the leading term. The term \(2x^2\) has power 2, so it is not the leading term. Exam tip: compare the powers of the variable before identifying the leading term.