Degree of a Polynomial

Correct answer: B. \(4x^2-5x+9\)

Explanation: The degree of a polynomial is the highest exponent of its variable with a non-zero coefficient. In \(4x^2-5x+9\), the highest power of \(x\) is \(2\), so its degree is \(2\). In contrast, \(3x^3+1\) has degree \(3\), and \(8\) is a constant polynomial with degree \(0\). Exam tip: simplify the polynomial first, then identify the greatest exponent of the variable.

Correct answer: C. 6

Explanation: The degree of a polynomial is the greatest exponent of the variable in a term with a non-zero coefficient. Here, the exponents of the terms are 6, 4, 1, and 0. In \(-2x^6\), the exponent of \(x\) is 6 and its coefficient, -2, is non-zero; therefore, the degree is 6. Although \(9x^4\) has degree 4, it is not the highest exponent. Exam tip: a non-zero constant term has degree 0.

Correct answer: D. 4

Explanation: The highest power of x in the polynomial is 4, appearing in the term 8x^4. Therefore, the degree of the polynomial is 4. The term -6x has degree 1 and the constant term 11 has degree 0, so neither determines the degree. Exam tip: To find the degree, identify the greatest exponent of the variable among terms with non-zero coefficients.

Correct answer: A. 2

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In the terms 15x^2, 9x, and -4, the powers of x are 2, 1, and 0 respectively. The greatest power is 2, so the degree is 2. Note that 15 is the coefficient of x^2, not the degree. Exam tip: To find the degree, look for the highest power of the variable, not the largest coefficient.

Correct answer: A. 0

Explanation: The polynomial 21 has no variable term; it can be written as 21x^0. Since 21 is a non-zero constant polynomial, its degree is 0. Its degree is not 1 because there is no term containing x to the power 1. Exam tip: Every non-zero constant polynomial has degree 0.

Correct answer: C. 3

Explanation: The coefficient of 0x^6 is 0, so it is a zero term and is not considered when finding the degree. The remaining polynomial is 5x^3-2, whose highest power of x is 3. Therefore, the correct answer is 3. Choosing 6 would be incorrect because the coefficient of x^6 is zero. Exam tip: Remove terms with zero coefficients before identifying the highest exponent.

Correct answer: D. 9

Explanation: The degree of a polynomial is the greatest exponent of the variable in any of its terms. Here, the exponents of x are 9, 2, and 1, and the greatest is 9. Therefore, the correct answer is 9. The number 7 is neither an exponent of x nor relevant to the degree. Exam tip: While finding degree, look only at the highest power of the variable, not at the coefficients.

Correct answer: A. 5

Explanation: In the polynomial 4 + 2x^5 - x^2, the highest power of x is 5, occurring in the term 2x^5. Therefore, the degree of the polynomial is 5. Although x^2 has exponent 2, the degree is determined by the greatest exponent. Exam tip: Regardless of the order of terms, identify the highest power of the variable.

Correct answer: B. 2

Explanation: The like terms 3x^3 and -3x^3 cancel each other. Therefore, the simplified polynomial is 8x^2-1. The highest exponent of x in it is 2, so its degree is 2. The term with exponent 3 does not remain after simplification, so the degree is not 3. Exam tip: Always simplify a polynomial by combining like terms before finding its degree.

Correct answer: C. \(-12x+4\)

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In \(-12x+4\), the highest exponent of \(x\) is \(1\), so its degree is \(1\). In contrast, \(7x^2+1\) has degree \(2\), while \(16\) is a constant polynomial of degree \(0\). Exam tip: identify the term with the highest power of the variable to find the degree quickly.

Correct answer: C. 8

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 8, 6, 2, and 0; the greatest is 8. Therefore, the degree is 8. The number 11 is only the coefficient of x^8, not the degree. Exam tip: To find degree, compare the exponents of the variable, not the coefficients.

Correct answer: A. 7x - 3

Explanation: The degree of a polynomial is the highest power of its variable. In 7x - 3, the highest power of x is 1, so it is a degree-one polynomial. The constant 5 has degree 0. Exam tip: ignore terms with zero coefficients when identifying degree.

Correct answer: C. 4

Explanation: The degree of a polynomial is the highest power of the variable in a term with a non-zero coefficient. Here, the term 2x^4 has power 4. The terms 0x^3 and 0x^2 are zero terms, so they do not affect the degree. The constant term 6 has power 0. Therefore, the degree of the polynomial is 4. Exam tip: Ignore terms with zero coefficients before finding the degree.

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

Explanation: The degree of a polynomial is the greatest exponent of the variable in any of its terms. In \(5x^3-2x+1\), the highest exponent of \(x\) is \(3\), so its degree is \(3\). The polynomial \(4x^2+7\) has degree \(2\), so it is not correct. Exam tip: a constant term has exponent \(0\).

Correct answer: B. 4

Explanation: Combining the like terms gives x^4+x^4=2x^4. Thus, the simplified polynomial is 2x^4+x^2. Its highest-power term with a non-zero coefficient is 2x^4, so the degree is 4. Although x^2 has degree 2, it is not the highest degree term. Exam tip: Simplify a polynomial first, then identify the greatest exponent among its non-zero terms.

Correct answer: A. 0

Explanation: (-33) is a non-zero constant polynomial because it has no variable. Every non-zero constant polynomial has degree 0, so the correct answer is 0. A polynomial of degree 1 must have a variable with highest power 1, which is absent here. Exam tip: the degree of any non-zero constant is always 0.

Correct answer: C. 1

Explanation: The terms 0x^5 and 0x^2 are zero, so they do not contribute to the polynomial. The expression becomes 9x-6, in which the highest power of x is 1. Therefore, its degree is 1. Choosing 5 is incorrect because the coefficient of x^5 is zero. Exam tip: Remove all terms with zero coefficients before finding the degree.

Correct answer: C. 6

Explanation: The degree of a polynomial is the greatest exponent of the variable in any term with a non-zero coefficient. In the given polynomial, the powers of x are 2, 6, 3, and 0. The greatest power is 6, so its degree is 6. Here, 10 is a coefficient, not an exponent, so it is not the degree. Exam tip: Look for the highest power of the variable, not the order of the terms.

Correct answer: C. -6

Explanation: -6 is a non-zero constant polynomial. Since it contains no variable with a positive exponent, its degree is 0. The degree of the zero polynomial is not defined, whereas the degrees of 5x and x^2+6 are 1 and 2 respectively. Exam tip: identify the highest exponent of the variable; every non-zero constant polynomial has degree 0.

Correct answer: B. 10

Explanation: The degree of a polynomial is the highest exponent of the variable having a non-zero coefficient. Here, the exponents of the terms are 10, 3, and 0, and the greatest is 10. Therefore, the correct answer is 10. Option 13 is the sum of exponents, not the degree. Exam tip: For degree, choose the greatest exponent; do not add the exponents.

Correct answer: C. the highest exponent of x, 3

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In the given polynomial, the powers of x are 3, 2, and 1, while the constant term has power 0. Hence, the greatest power is 3, so the degree is 3. Although 7 is the largest coefficient, it does not determine the degree. Exam tip: simplify the polynomial first, then identify the highest power of the variable.

Correct answer: B. 1

Explanation: Combining like terms gives \(2x^2-2x^2=0\). Thus, the polynomial becomes \(4x-3\). The highest power of \(x\) in this simplified polynomial is 1, so its degree is 1. Option 2 may seem tempting if the original \(x^2\) terms are considered without simplifying, but those terms cancel out. Exam tip: Always simplify a polynomial before finding its degree.

Correct answer: A. 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 the terms are 8, 5, 2, and 0. The greatest exponent is 8, so the correct answer is 8. Option 5 is only the exponent of the second term, not the degree of the polynomial. Exam tip: Treat a non-zero constant term as having exponent 0, then identify the greatest exponent.

Correct answer: C. \(6x^3+4\)

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In \(6x^3+4\), the highest power of \(x\) is \(3\), so its degree is \(3\), not \(4\). In contrast, \(x^4+1\), \(5x^4-2x\), and \(9x^4+x^2\) each contain an \(x^4\) term, so each has degree \(4\). Exam tip: To find the degree, identify the highest exponent of the variable.

Correct answer: C. 2

Explanation: Here, \(x^0=1\), so the polynomial becomes \(1+5x^2-8=5x^2-7\). The highest exponent with a non-zero coefficient is \(2\); therefore, its degree is \(2\). The exponent \(0\) may occur in a constant term, but it is not the degree here. Exam tip: simplify like terms first, then identify the greatest exponent with a non-zero coefficient.