Degree of a Polynomial

Correct answer: D. Not defined

Explanation: The given expression simplifies to 0x^3+0x+0 = 0, so it is the zero polynomial. Since the zero polynomial has no non-zero term, its degree is not defined. It cannot be 3, 1, or 0 because the coefficients of all those powers are zero. Exam tip: simplify a polynomial by removing zero-coefficient terms before finding its degree.

Correct answer: B. 7

Explanation: A term with coefficient zero does not contribute to the degree of a polynomial. Here, \(0x^9=0\), so the polynomial becomes \(12x^7-5\). Its highest power with a non-zero coefficient is \(7\); therefore, its degree is 7. Option 9 is incorrect because the coefficient of \(x^9\) is zero. Exam tip: First remove all zero-coefficient terms before finding the degree.

Correct answer: D. 4

Explanation: The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. The given polynomial has the terms x, x^2, x^3 and x^4, and the greatest exponent of x is 4. Hence, its degree is 4. Option 3 is not correct because the term x^4 is also present. Exam tip: First simplify a polynomial, then identify the greatest exponent of the variable.

Correct answer: A. \(4x^5-3x^2+1\)

Explanation: The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. In \(4x^5-3x^2+1\), the highest power of \(x\) is \(5\), so its degree is \(5\). Option B has highest power \(4\), so it is not correct. Exam tip: To find the degree, look for the greatest exponent of the variable, not the constant term.

Correct answer: B. It is the highest exponent of the variable whose coefficient is non-zero.

Explanation: The degree of a non-zero polynomial is the greatest exponent with a non-zero coefficient. For example, in 5x^4-2x, it is 4, not the number of terms. Exam tip: ignore any term whose coefficient is zero.

Correct answer: C. 6

Explanation: The degree of a polynomial is the highest exponent of the variable among terms with non-zero coefficients. Here, the exponents of x are 2, 6, and 4. The greatest exponent is 6, so the degree of the polynomial is 6. Note that 7 is the coefficient of x^6, not the degree. Exam tip: Identify the exponent in each term first, then select the greatest one.

Correct answer: C. 6

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. Since the highest power given is 6, the degree of the polynomial is 6. Degree 0 applies only to a non-zero constant polynomial, not to a polynomial whose highest power is 6. Exam tip: Ignore any terms with zero coefficients before identifying the highest remaining power.

Correct answer: C. 4

Explanation: In \(7x^4-3x^2+5\), the powers of \(x\) are 4, 2, and 0. The greatest exponent of a term with a non-zero coefficient is 4, so the degree is 4. The number \(7\) is only the coefficient of \(x^4\), not the degree. Exam tip: simplify like terms first, then identify the highest exponent.

Correct answer: A. The degree is 5 because the highest power of \(x\) is 5.

Explanation: The degree of a polynomial is the greatest exponent of the variable in any term. Here the exponents are 2, 5, and 0, so the degree is 5. The first term is irrelevant. Exam tip: check every term’s power.

Correct answer: C. 6

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. The given polynomial has the terms 5x^6, 2x^3, -x, and 8, and the greatest power of x is 6. Hence, its degree is 6. The constant term 8 has degree 0, so it does not determine the degree. Exam tip: Compare the exponents of the variable in all terms before choosing the degree.

Correct answer: C. 2

Explanation: The coefficient of \(0x^5\) is 0, so it is a zero term and is not considered when finding the degree. The remaining polynomial is \(4x^2-9x+6\), whose highest power of \(x\) is 2. Therefore, its degree is 2. Option 5 is a close distractor, but the \(x^5\) term has coefficient 0 and does not contribute to the degree. Exam tip: Remove all zero-coefficient terms before identifying the highest exponent.

Correct answer: A. 2x⁵

Explanation: The governing concept is the degree of a polynomial: for a non-zero polynomial in one variable, its degree is the greatest exponent of the variable having a non-zero coefficient. In 2x⁵ + 9x³ − 4, the powers of x are 5, 3 and 0 respectively. The largest of these is 5, and the corresponding term is 2x⁵. Therefore, option A is correct. The coefficient 2 does not determine the degree; it only gives the numerical coefficient of the leading term. The term 9x³ has a smaller exponent, the constant term −4 has degree zero, and 9 alone is not a term of the given polynomial. Thus the highest-power term, not the largest coefficient or the last term written, determines the degree.

Correct answer: C. 8

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 x are 3, 8, and 2, and the greatest is 8. Therefore, the degree of the polynomial is 8. Option 13 is incorrect because exponents of separate terms are not added to find the degree. Exam tip: Rearranging the terms does not change the degree of a polynomial.

Correct answer: C. 1

Explanation: On combining like terms, \(4x^4-4x^4=0\). Thus, the simplified polynomial is \(7x-2\). The highest power of \(x\) in it is 1, so its degree is 1. Choosing 4 is incorrect because the \(x^4\) terms cancel each other. Exam tip: Always simplify a polynomial before finding its degree.

Correct answer: B. 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. Since the coefficient of -9x^7 is -9, which is non-zero, the highest exponent is 7. Therefore, the degree is 7. Option 4 is only the exponent of another term, and 9 is not an exponent in the polynomial. Exam tip: A non-zero constant term such as 1 has degree 0.

Correct answer: C. 3

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 3, 2, and 0, so the greatest exponent is 3. Therefore, the correct answer is 3. The number 6 is the coefficient of x^3, not the degree. Exam tip: First simplify the polynomial, then identify the highest power of the variable.

Correct answer: B. 1

Explanation: In the polynomial \(12x-7\), the variable term is \(12x=12x^1\). The highest exponent of \(x\) is \(1\), so the degree of the polynomial is \(1\). The constant term \(-7\) has degree 0 and does not determine the degree here. Exam tip: the degree of a polynomial is the highest exponent of its variable.

Correct answer: C. 6

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 6, 4, 1, and 0. The greatest exponent is 6, so the degree of the polynomial is 6. Option 4 is a close distractor because it is the exponent of the second term, not the highest exponent. Exam tip: Simplify a polynomial first and ignore any terms whose coefficients become zero.

Correct answer: A. 0

Explanation: The degree of a non-zero constant polynomial is 0 because \(27\) can be written as \(27x^0\), and the highest power of \(x\) is 0. Here, 27 is the value (or coefficient), not the degree. Exam tip: only the zero polynomial has an undefined degree.

Correct answer: C. 2

Explanation: The coefficient of \(0x^5\) is zero, so it is not considered an actual term of the polynomial. The remaining polynomial is \(8x^2-3x+4\), whose highest power of \(x\) is \(2\). Therefore, its degree is \(2\). It is not \(5\) because the coefficient of \(x^5\) is zero. Exam tip: Remove all terms with zero coefficients before finding the degree.

Correct answer: A. 3x⁴

Explanation: The governing concept is the degree of a polynomial. For a non-zero polynomial, the degree is the greatest exponent of the variable among terms with non-zero coefficients. In 3x⁴ + 7x² − 5, the exponents of x are 4, 2 and 0. Since 4 is the greatest exponent, the term containing it is 3x⁴, so option A is correct. The coefficient 3 affects the value of the term but does not change its degree. The term 7x² has degree 2, the constant term −5 has degree 0, and 7 by itself is not a complete term appearing in the polynomial. Consequently, neither the largest coefficient nor the constant term determines the degree; the highest power of x does.

Correct answer: D. 7

Explanation: The degree of a polynomial is the greatest exponent of the variable among its terms with non-zero coefficients. Here, the exponents are 2, 7, and 1. The greatest exponent is 7, so the degree of the polynomial is 7. Option 2 is only the exponent of the first term, x^2, not of the whole polynomial. Exam tip: Ignore the order of terms and identify the highest exponent of the variable.

Correct answer: C. 1

Explanation: On simplification, the like terms cancel: \(6x^3-6x^3=0\). The polynomial left is \(9x-2\). The highest power of \(x\) in it is \(1\), so its degree is \(1\). Degree 0 would apply only to a non-zero constant polynomial, but \(9x\) is present here. Exam tip: simplify like terms first, then identify the highest exponent of the variable remaining.

Correct answer: C. 8

Explanation: The degree of a polynomial is the greatest exponent of the variable whose coefficient is non-zero. Here, the exponents of the terms are 8, 5, 2, and 0. The greatest exponent is 8, and the coefficient of x^8 is -4, which is non-zero. Therefore, the degree of the polynomial is 8. The constant term 6 has exponent 0, so it does not determine the degree. Exam tip: identify the highest non-zero power of the variable.

Correct answer: B. 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; the greatest is 4. Therefore, the degree is 4. The number 9 is the coefficient of x^4, not the degree. Exam tip: Treat a non-zero constant term as having exponent 0 and identify the highest exponent of the variable.