Degree of a Polynomial

Correct answer: B. 4

Explanation: The coefficient of \(0x^{10}\) is 0, so it is not considered a non-zero term of the polynomial. Among the remaining terms, \(3x^4\) has the highest power, 4; therefore, the degree is 4. Choosing 10 would be incorrect because the coefficient of \(x^{10}\) is zero. Exam tip: To find the degree, look for the highest exponent with a non-zero coefficient.

Correct answer: C. 2

Explanation: On simplification, the like terms 9x^6 and -9x^6 cancel each other. The polynomial left is 5x^2+1. The highest power of x in it is 2, so its degree is 2. Option 6 is incorrect because the x^6 term does not remain after simplification. Exam tip: Always simplify a polynomial by combining like terms before finding its degree.

Correct answer: B. \(4x^7-x^2+5\)

Explanation: The degree of a polynomial is the highest power of its variable with a non-zero coefficient. In \(4x^7-x^2+5\), the highest power of \(x\) is 7, so its degree is 7. \(2x^6+3\) has degree 6, whereas \(19\) is a constant polynomial with degree 0. Exam tip: First write the polynomial in simplified form, then identify the highest exponent of the variable.

Correct answer: C. 3

Explanation: Here, x^0=1, so the polynomial can be written as 1+2x^3-8x. The highest exponent of x with a non-zero coefficient is 3; therefore, its degree is 3. The number 8 is a coefficient, not an exponent. Exam tip: Simplify the polynomial first, then identify the highest power of x.

Correct answer: D. 8

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 1, 4, and 8, while the constant term 6 has degree 0. The greatest power is 8, so the degree of the polynomial is 8. The number 6 is a constant term, not the degree. Exam tip: Ignore the order of terms and identify the highest power of the variable.

Correct answer: A. \(0\)

Explanation: \((-25)\) is a non-zero constant polynomial because it has no variable term. Every non-zero constant polynomial has degree \(0\). A polynomial of degree \(1\) must have a variable with highest power \(1\), which is absent here. Exam tip: only the zero polynomial has an undefined degree; every non-zero constant has degree \(0\).

Correct answer: D. Not defined

Explanation: The given expression 0x² + 0x + 0 equals 0, so it is the zero polynomial. Since the zero polynomial has no non-zero term, its degree is not defined. Degree 0 applies to a non-zero constant polynomial such as 5, not to the zero polynomial. Exam tip: while finding degree, consider only terms with non-zero coefficients.

Correct answer: B. 4

Explanation: The degree of a polynomial is the greatest exponent of the variable in any of its terms. Here, the exponents are 4, 2, and 0, and the greatest is 4. Therefore, the degree of the polynomial is 4. The exponent 2 belongs only to the term 5x^2. Exam tip: A non-zero constant term has degree 0.

Correct answer: B. 1

Explanation: In the polynomial 13x+8, the term 13x has x raised to the power 1, while the constant term 8 has degree 0. Since the highest power of x is 1, the degree of the polynomial is 1. The numbers 13 and 8 are coefficients/constants, not the degree. Exam tip: Find the highest exponent of the variable to determine the degree of a polynomial.

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

Explanation: A quadratic polynomial has highest exponent 2. In \(3x^2-x+7\), the greatest power of \(x\) is 2, so it is quadratic. \(x^3+4x\) has degree 3. Exam tip: determine degree from the highest non-zero power only.

Correct answer: A. Correct; the degree of a polynomial is the highest power of the variable in it.

Explanation: The statement is correct. The powers of the terms are 3, 1, and 0, so the greatest power is 3 and the degree is 3. Coefficients and the number of terms do not decide degree. Exam tip: check the highest exponent first.

Correct answer: C. 3

Explanation: The term \(0x^7\) has coefficient zero, so it is not considered an actual term of the polynomial. In the remaining polynomial \(6x^3-2x+5\), the highest power of \(x\) is \(3\). Therefore, its degree is \(3\). Choosing \(7\) would be incorrect because a term with zero coefficient does not determine the degree. Exam tip: Remove all zero-coefficient terms before finding the degree.

Correct answer: A. 5x⁶

Explanation: For a non-zero polynomial in one variable, the degree is the greatest exponent of the variable whose coefficient is non-zero. In 5x^6 + 2x^4 − 11, the powers of x are 6, 4, and 0, because the constant term −11 can be written as −11x^0. The greatest exponent is 6, and it occurs in 5x^6, whose coefficient 5 is non-zero. Thus option A is correct and the polynomial has degree 6. The coefficient 5 does not determine the degree; it only multiplies the power of x. The term 2x^4 has a smaller exponent, −11 has degree zero, and the isolated number 2 is not a term in the displayed polynomial. The leading term therefore determines the answer.

Correct answer: C. 9

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, 9, and 2. The greatest exponent is 9, so the degree of the polynomial is 9. Option 4 is only the exponent of the first term, \(x^4\), not the degree of the whole polynomial. Exam tip: Ignore the order of terms and identify the highest exponent of the variable.

Correct answer: B. 2

Explanation: Here, 8x^3 and -8x^3 are opposite like terms, so they cancel to 0. The simplified polynomial is 4x^2-1. The highest power of x in it is 2, so its degree is 2. It is not degree 3 because the x^3 terms have cancelled. Exam tip: Always simplify by combining like terms before finding the degree of a polynomial.

Correct answer: A. बहुपद की डिग्री शून्येतर गुणांक वाले चर के सबसे बड़े घातांक के बराबर होती है।

Explanation: For \(4x^5-2x+9\), the exponents are 5, 1, and 0; the greatest is 5, so its degree is 5. It has three terms, but term count is not degree. In exams, identify the term with the highest exponent.

Correct answer: B. 3

Explanation: In \(10x^3-9x+14\), the highest exponent of \(x\) is \(3\), in the term \(10x^3\). Therefore, the degree of the polynomial is \(3\). Here, \(10\) is a coefficient, not the degree, and \(14\) is a constant term. Exam tip: To find the degree, identify the greatest exponent of the variable, not the coefficient.

Correct answer: A. 2

Explanation: The degree of a polynomial is the highest exponent of its variable. In the terms 16x^2, 5x, and -3, the powers of x are 2, 1, and 0 respectively. Hence, the highest power is 2, so the degree of the polynomial is 2. The number 16 is a coefficient, not the degree. Exam tip: For degree, look at the greatest power of the variable, not its coefficient.

Correct answer: A. 0

Explanation: 38 is a non-zero constant polynomial because it has no variable. The degree of every non-zero constant polynomial is 0, so 0 is the correct answer. A polynomial of degree 1 must have a variable with highest power 1, such as \(x+38\). Exam tip: Remember that every non-zero constant polynomial has degree 0.

Correct answer: C. 4

Explanation: The coefficient of \(0x^8\) is zero, so it does not contribute to the polynomial. In the remaining polynomial \(9x^4-5\), the highest power of \(x\) is \(4\); therefore, its degree is \(4\). Option 8 is incorrect because the coefficient of \(x^8\) is zero. Exam tip: Remove terms with zero coefficients before finding the degree.

Correct answer: C. 12

Explanation: The degree of a polynomial in one variable is the greatest exponent of the variable with a non-zero coefficient. Here, the exponents of x are 12, 5, and 1, and the greatest is 12. Therefore, the degree of the polynomial is 12. The number 5 is only the exponent in the term 4x^5, not the degree of the whole polynomial. Exam tip: Compare the exponents of all terms and select the greatest one.

Correct answer: A. 6

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. Here, the exponents of the terms are 0, 6, and 3, and the greatest is 6. Therefore, the degree of the polynomial is 6. The constant term 7 has degree 0. Exam tip: The order of terms does not matter; identify the highest exponent of the variable.

Correct answer: B. 3

Explanation: The like terms 5x^5 and -5x^5 cancel each other. The simplified polynomial is 6x^3+2. The highest power of x in it is 3, so its degree is 3. Although 5 appears in the original expression, its terms cancel and therefore it does not determine the degree. Exam tip: Always simplify a polynomial before finding its degree.

Correct answer: A. 8x^4-3x+1

Explanation: The degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. In option A, the highest power of x is 4, so its degree is 4. In option B, the highest power is 3, so it is not correct. Exam tip: First write the polynomial in simplified form, then identify the highest power of the variable.

Correct answer: B. 6

Explanation: Combining like terms gives \(x^6+x^6+x^2=2x^6+x^2\). The highest power of \(x\) in this polynomial is 6, so its degree is 6. It is not 8 or 12, because the degree is found from the highest exponent in the simplified polynomial, not by adding exponents. Exam tip: simplify like terms first, then identify the greatest exponent with a non-zero coefficient.