Definition of a Polynomial

Correct answer: A. 4, 2, 0

Explanation: The governing rule is that the power of a term is the exponent of the variable, while a nonzero constant is treated as having power 0. Thus 2x⁴ contributes power 4, 5x² contributes power 2, and 1 contributes power 0. The coefficient numbers 2 and 5 are not powers, so option A is correct; the other choices confuse coefficients or introduce powers not present.

Correct answer: C. 5

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 5, 3, 1, and 0. The greatest exponent is 5, so the degree of the polynomial is 5. The number 6 is a coefficient, not the degree. Exam tip: To find the degree, look for the highest power of the variable.

Correct answer: A. Linear polynomial

Explanation: In \(8x-11\), the highest power of \(x\) is \(1\), so its degree is \(1\). A polynomial with 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 power of the variable.

Correct answer: B. Quadratic polynomial

Explanation: The highest power of x is 2 because the term 7x^2 is present. Therefore, the degree of this polynomial is 2, so it is a quadratic polynomial. A linear polynomial has degree 1, but this expression also contains an x^2 term. Exam tip: To identify the type of a polynomial, look for the highest exponent 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 \(3x^3-5x+2\), the highest power of \(x\) is 3, so it is a cubic polynomial. The terms \(-5x\) and 2 have degrees 1 and 0 respectively, so they do not affect the degree. Exam tip: To identify the type of a polynomial, first locate the highest power of the variable.

Correct answer: D. Constant polynomial

Explanation: The expression 21 contains no variable, such as x, so its value always remains 21. Therefore, it is a non-zero constant polynomial and has degree 0. A zero polynomial has all its coefficients equal to 0, whereas the constant here is 21. Exam tip: An expression containing only a non-zero number is a constant polynomial.

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

Explanation: In a polynomial, every exponent of a variable must be a non-negative integer. Here, the exponent of \(x\) is \(\frac{7}{3}\), which is fractional, so \(x^{\frac{7}{3}}+2\) is not a polynomial. A constant term such as 2, addition, and having two terms do not prevent an expression from being a polynomial. Exam tip: An expression is not a polynomial if a variable has a fractional or negative exponent, or appears in an exponent.

Correct answer: A. क्योंकि x की एक घात ऋणात्मक है

Explanation: In a polynomial, the exponent of a variable must be 0 or a positive integer. This expression contains x^{-5}, where the exponent of x is -5, so it is not a polynomial. The term 9x is a valid linear term; having two terms or an addition sign does not prevent an expression from being a polynomial. Exam tip: An expression with a negative, fractional, or variable exponent is not a polynomial.

Correct answer: D. 7

Explanation: The degree of a polynomial is the greatest exponent of the variable among terms with non-zero coefficients. Here, the exponents are 7, 4, 2, and 0. The greatest exponent is 7, so the degree of the polynomial is 7. The term -6 is a constant term and has degree 0. Exam tip: To find the degree, look for the highest power of the variable, not the coefficient.

Correct answer: C. 4

Explanation: Terms of a polynomial are algebraic parts separated by addition or subtraction. Here the terms are \(5x^4\), \(-3x^2\), \(x\), and \(-12\). Therefore, there are 4 terms. Note that \(-3x^2\) is one complete term; the negative sign is part of its coefficient. Exam tip: Count each complete algebraic part separated by + or − as one term.

Correct answer: A. \(9x^5\)

Explanation: The degrees of the terms are \(5\), \(3\), and \(0\), respectively. The greatest power is \(5\), which occurs in the term \(9x^5\). The term \(x^3\) has power \(3\), so it is not the highest. Exam tip: Find a term’s degree by looking at the exponent of its variable; a constant term has degree \(0\).

Correct answer: A. Polynomial

Explanation: This is a polynomial because the powers of the variable \(x\) are \(2\), \(1\), and \(0\), all of which are non-negative integers. Although \(\sqrt{7}\) is irrational, it is a real number and can be a valid coefficient of \(x\). The presence of \(\sqrt{7}\) does not make the expression a non-polynomial. Exam tip: In a polynomial, variable exponents must be \(0,1,2,\ldots\), while coefficients may be any real numbers.

Correct answer: C. Not a polynomial

Explanation: The correct answer is not a polynomial. Since \(\sqrt{x}=x^{\frac{1}{2}}\), the exponent of \(x\) in the expression is \(\frac{1}{2}\). In a polynomial, variable exponents must be zero or positive integers; fractional exponents are not allowed. Therefore, it is neither a linear nor a quadratic polynomial. Exam tip: Rewrite a square root as an exponent and check the exponent of the variable.

Correct answer: C. 0

Explanation: A constant term contains no variable; equivalently, it has \(x\) raised to the power 0. The expression \(14x^2\) has only an \(x^2\) term and no term without a variable. Therefore, its constant term is \(0\). Here, \(14\) 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 polynomial \(x^6+x^2+1\) contains only the terms \(x^6\), \(x^2\), and the constant term \(1\); it has no \(x^4\) term. When a term of a particular power is absent, its coefficient is taken as \(0\). Therefore, the correct answer is \(0\). Note that \(1\) is the coefficient of the constant term, not of \(x^4\). Exam tip: First locate the term with the exact power asked in the question.

Correct answer: A. Yes, because \(x^0=1\), so the expression becomes \(3+5x\), in which the exponents of \(x\) are non-negative integers.

Explanation: Yes. Since \(x^0=1\), \(3x^0+5x=3+5x\). The exponents of \(x\) are 0 and 1, both non-negative integers, so it is a polynomial. Option B is incorrect because an exponent of zero is allowed in a polynomial. Exam tip: When you see \(x^0\), simplify it to 1 before deciding whether the expression is a polynomial.

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

Explanation: In (\frac{5x}{7}), the coefficient is (\frac{5}{7}) and the power of (x) is (1). A fractional coefficient is valid in a polynomial.

Correct answer: A. Yes, because coefficients can be real and powers are valid

Explanation: The direct answer is A: yes, it is a polynomial. A polynomial is an expression made from numbers and a variable in which every variable exponent is a non-negative integer: 0, 1, 2, 3, and so on. In this expression, the terms are \\(-\\frac{4}{9}x^4\\), \\(2x\\), and \\(-1\\). Their powers of x are 4, 1, and 0, respectively. The coefficients \\(-\\frac{4}{9}\\), 2, and \\(-1\\) are allowed; coefficients may be negative, fractional, or zero. Therefore the expression satisfies the definition. Option A is correct because all powers are valid and the coefficients are real numbers. Option B is wrong because a negative fraction is still a valid coefficient. Option C is wrong because \\(x^4\\) is perfectly acceptable; 4 is a non-negative integer. Option D is wrong because having three terms does not prevent an expression from being a polynomial; a polynomial may have one, two, three, or many terms. Remember: check the powers of the variable, not whether coefficients are negative or fractional and not merely how many terms appear.

Correct answer: B. \(x^{\frac{5}{6}}+4\)

Explanation: In a polynomial, every exponent of a variable must be a non-negative integer, such as \(0,1,2,\ldots\). In \(x^{\frac{5}{6}}+4\), the exponent of \(x\) is fractional, so it is not a polynomial. In contrast, \(x^5+6\) has exponent 5 and is a polynomial. Exam tip: while identifying polynomials, check first for negative or fractional exponents of variables.

Correct answer: C. 2

Explanation: The degree of a polynomial is the highest power of its variable with a non-zero coefficient. Here, the terms are 8x^2, -9x, and 16, and the highest power of x is 2. Therefore, the degree is 2. The number 8 is a coefficient, not the degree. Exam tip: Look for the highest exponent, not the largest coefficient.

Correct answer: B. 1

Explanation: In the polynomial 17x - 5, the power of x in the term 17x is 1 because x = x^1. The constant term -5 has power 0. Since the highest power is 1, its degree is 1. The numbers 5 and 17 are the constant term and coefficient respectively, not the degree. Exam tip: To find the degree, identify the highest exponent of the variable.

Correct answer: A. \(0\)

Explanation: A non-zero constant polynomial has no variable term. It can be written as \(-31x^0\), so the highest power of \(x\) is \(0\); hence its degree is \(0\). A polynomial of degree \(1\) is linear, such as \(x-31\), so it is not correct here. Exam tip: only the zero polynomial has an undefined degree; every non-zero constant has degree \(0\).

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

Explanation: The degree of a polynomial is determined by the greatest exponent whose coefficient is not zero. A non-zero constant, such as 7 or -2, has degree 0 because it can be written as a constant multiplied by \\(x^0\\). The zero polynomial is not the same as an ordinary constant: every coefficient is zero, so there is no greatest exponent with a non-zero coefficient.

Hence the degree of the zero polynomial is not defined, while the degree of a non-zero constant polynomial is 0. This makes option C correct. Option A overlooks the special nature of the zero polynomial. Option B assigns an unsupported degree, and option D is false because the zero polynomial is included among polynomials, even though its degree requires special treatment.

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, \(-5x^6\) has exponent 6, which is greater than the exponents of \(3x^4\), \(-x\), and the constant \(9\), namely 4, 1, and 0. Therefore, \(-5x^6\) is the leading term. Exam tip: Compare exponents; whether the coefficient is positive or negative does not determine the leading term.

Correct answer: A. \(4x^3-7x+1\)

Explanation: In \(4x^3-7x+1\), the powers of x are 3, 1 and 0, all non-negative integers, so it is a polynomial. But \(3/x=3x^{-1}\), which is not allowed. Exam tip: reject expressions with negative or fractional powers.