Degree of a Polynomial

Correct answer: C. 1

Explanation: In the given expression, the coefficients of \(0x^5\) and \(0x^3\) are zero, so these terms do not contribute to the polynomial. The remaining polynomial is \(12x-4\). The highest power of \(x\) in it is \(1\), so its degree is \(1\). Degree \(0\) applies to a non-zero constant polynomial, not to this linear polynomial. Exam tip: Remove all terms with zero coefficients before finding the degree.

Correct answer: C. 8

Explanation: The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. In the given polynomial, the exponents are 2, 8, 4, and 0. The greatest exponent is 8, so its degree is 8. Here, 14 is a coefficient, not the degree. Exam tip: Arrange terms in descending powers to identify the degree quickly.

Correct answer: B. 13

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 13, 6, and 0, and the greatest is 13. Therefore, the correct answer is 13. The number 6 is only the exponent of the second term, not the degree. Exam tip: Treat a non-zero constant term as having exponent 0 and select the highest exponent.

Correct answer: C. Highest power (3)

Explanation: The degree of a polynomial in one variable is the greatest power of the variable with a non-zero coefficient. Here, the powers of x are 3, 2, 1 and 0; the highest power is 3, so the polynomial has degree 3. Although 9 is the largest coefficient and 20 is the constant term, neither determines the degree. Exam tip: identify the greatest exponent of the variable after writing the polynomial in standard form.

Correct answer: B. 1

Explanation: The like terms 7x^2 and -7x^2 cancel each other. The simplified polynomial is therefore 10x-5. The highest power of x in it is 1, so its degree is 1. It is not 2 because the x^2 terms become zero after simplification. Exam tip: Always simplify a polynomial by combining like terms before finding its degree.

Correct answer: A. 9

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. Here, the powers of x are 9, 6, and 3, while the constant term −7 has degree 0. The greatest power is 9, so the degree is 9. Option 6 is only the exponent in the term −8x^6, not the degree of the whole polynomial. Exam tip: list the exponents of the variable and select the greatest one.

Correct answer: C. \(7x^4+5\)

Explanation: The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. In \(7x^4+5\), the highest power of \(x\) is \(4\), so its degree is \(4\), not \(5\). Each of the other three polynomials contains an \(x^5\) term and therefore has degree \(5\). Exam tip: To find the degree, look for the highest power of the variable; a constant term does not affect it.

Correct answer: C. 3

Explanation: Here, \(x^0=1\), so the polynomial becomes \(1+11x^3-6=11x^3-5\). Its term with the highest exponent is \(11x^3\), so the degree of the polynomial is \(3\). The number \(11\) is a coefficient, not an exponent. Exam tip: Simplify terms such as \(x^0\) first, then identify the greatest exponent of the variable.

Correct answer: D. Not defined

Explanation: On simplifying, the given expression becomes 0, so it is the zero polynomial. The degree of a non-zero polynomial is the highest power of a term with a non-zero coefficient, but the zero polynomial has no such term. Hence, its degree is not defined. Degree 0 applies to a non-zero constant polynomial, such as 5. Exam tip: simplify a polynomial and remove terms with zero coefficients before finding its degree.

Correct answer: B. 7

Explanation: The term \(0x^{11}\) has zero coefficient, so it is not an actual term of the polynomial. Among the remaining non-zero terms, \(18x^7\) and \(-2\), the greatest power of \(x\) is \(7\). Therefore, the degree of the polynomial is \(7\). Option 11 is tempting, but a term with coefficient zero does not affect the degree. Exam tip: Remove all zero-coefficient terms before finding the degree.

Correct answer: D. 6

Explanation: The degree of a polynomial is the greatest exponent of the variable among its terms with non-zero coefficients. Here, the exponents are 1, 2, 4, and 6. The greatest exponent is 6, so the degree of the polynomial is 6. Option 4 may seem close, but the term x^6 is present, so the degree cannot be 4. Exam tip: identify the highest power of the variable in the polynomial.

Correct answer: A. \(4x^7-3x^2+1\)

Explanation: The degree of a polynomial is the highest power of its variable with a non-zero coefficient. In \(4x^7-3x^2+1\), the highest power of \(x\) is \(7\), so its degree is \(7\). In option B, the highest power is \(6\), so it is not correct. Exam tip: identify the greatest exponent of the variable in the polynomial.

Correct answer: C. Reema is incorrect; the degree is 2 because the highest exponent of \(p\) is 2.

Explanation: The degree of a polynomial is the highest exponent of its variable, not the number of terms. Here the exponents in \(6p^2\), \(-p\), and 8 are 2, 1, and 0, so the degree is 2. Exam tip: identify the highest power first.

Correct answer: C. 8

Explanation: The degree of a polynomial is the greatest exponent of its variable. The exponents in the given polynomial are 2, 8, and 4, and the greatest is 8. Therefore, its degree is 8. The number 6 is the coefficient of x^8, not the degree. Exam tip: identify the highest power of the variable, regardless of the order of terms.

Correct answer: C. 10

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. Since the highest power given is 10, the degree of the polynomial is 10. Degrees 0 and 1 apply only to constant and linear polynomials, respectively. Exam tip: Ignore terms with zero coefficients and identify the greatest remaining exponent.

Correct answer: C. 4

Explanation: Combining like terms gives 3x^4+8x^4=11x^4. Therefore, the simplified polynomial is 11x^4-2x+1. The highest power of x is 4, so its degree is 4. Here, 11 is a coefficient, not the degree. Exam tip: Simplify the polynomial first, then identify the highest exponent of the variable.

Correct answer: C. 6

Explanation: The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. The exponents of the terms in the given polynomial are 3, 0, and 6; the greatest is 6. Therefore, the correct answer is 6. Option 3 is only the degree of the term 5x^3, not of the whole polynomial. Exam tip: Ignore the order of terms and identify the highest power of the variable.

Correct answer: B. 5

Explanation: The coefficient of \(0x^{12}\) is 0, so it is a zero term and is ignored while finding the degree. Among the remaining non-zero terms, \(4x^5\), \(-3x^2\), and \(8\), the highest exponent is 5. Therefore, the degree of the polynomial is 5. Choosing 12 would be incorrect because the coefficient of \(x^{12}\) is zero. Exam tip: Remove all terms with zero coefficients before finding the degree.

Correct answer: B. 3

Explanation: On subtracting the like terms, \(11x^7-11x^7=0\). Thus, the simplified polynomial is \(3x^3+2\). The highest power of \(x\) is 3, so its degree is 3. Option 7 is incorrect because the \(x^7\) terms cancel out. Exam tip: Always simplify a polynomial before finding its degree.

Correct answer: B. \(4x^8-x^2+5\)

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In \(4x^8-x^2+5\), the highest power of \(x\) is \(8\), so its degree is \(8\). \(2x^7+3\) has degree \(7\), so it is not correct. Exam tip: a constant term has exponent \(0\).

Correct answer: C. 5

Explanation: Since \(x^0=1\), the polynomial becomes \(1+4x^5-9x\). The highest exponent of \(x\) with a non-zero coefficient is 5, so its degree is 5. Option 9 is a distractor based on the coefficient; degree depends on the exponent, not on the coefficient. Exam tip: simplify the polynomial first, then identify the greatest exponent among terms with non-zero coefficients.

Correct answer: D. 9

Explanation: The degree of a polynomial is the greatest exponent of the variable among terms with non-zero coefficients. Here, the powers of x are 1, 4, and 9, while the constant term 6 has power 0. The greatest exponent is 9, so the degree of the polynomial is 9. The number 6 is a constant term, not the degree. Exam tip: The order of terms does not matter; identify the highest power of the variable.

Correct answer: A. 0

Explanation: (-52) is a non-zero constant polynomial because it has no variable term. Every non-zero constant polynomial has degree 0. Degree 1 is for a linear polynomial, whose highest power of a variable is 1. Exam tip: The degree of any non-zero constant is always 0.

Correct answer: B. 6

Explanation: The term \(0x^{14}\) has coefficient 0, so it is a zero term and is not considered while finding the degree. The remaining polynomial is \(6x^6-2x+3\), whose highest power of \(x\) is 6. Hence, its degree is 6. Option 14 is a close distractor, but that term does not contribute because its coefficient is 0. Exam tip: Remove terms with zero coefficients before identifying the highest exponent.

Correct answer: A. \(4x^3-2x+7\)

Explanation: In a cubic polynomial, the highest exponent of the variable is 3. In \(4x^3-2x+7\), the highest power is 3, so it is cubic. Option B is quadratic. Exam tip: identify the degree from the highest power in simplified form.