Degree of a Polynomial

Correct answer: A. The degree is 5 because the highest exponent of a term with a non-zero coefficient is 5.

Explanation: The degree is the greatest exponent among terms having non-zero coefficients. Here the exponents are 2, 5, and 0, so the degree is 5. The first written term does not decide it. Exam tip: scan every term and select the highest power of the variable.

Correct answer: B. 11

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 x are 11, 8, 4, and 0; the greatest is 11. Therefore, the degree of the polynomial is 11. The number 4 is only the coefficient of x^{11}, while 8 is the exponent in another term. Exam tip: Treat a constant term as having exponent 0 and identify the highest exponent.

Correct answer: C. 9

Explanation: The degree of a polynomial is the greatest exponent of the variable whose coefficient is non-zero. Since n > 4, the exponent n in x^n is greater than 4, so it must be the highest exponent. As the degree is given as 9, n = 9. Option 4 is not possible because the question states n > 4. Exam tip: while finding degree, check that the coefficient of the highest-power term is not zero.

Correct answer: A. (a) can be any real number

Explanation: A polynomial’s degree is controlled by its highest power with a non-zero coefficient. In x^5+ax^3+4, the coefficient of x^5 is 1, so the fifth-degree term cannot disappear for any value of a. Changing a only changes the coefficient of the lower-power term x^3. It cannot affect the leading term or reduce the degree.

For example, when a=0, the polynomial is x^5+4 and its degree is still 5. When a is any other real number, the x^3 term is present, but x^5 is still the highest power. Hence every real value of a satisfies the stated degree condition. Option A is correct. Neither a=5 nor a≠0 is required; in particular, a=0 is allowed.

Correct answer: D. 0

Explanation: The governing rule is that zero-coefficient terms do not contribute to a polynomial and must be removed before its degree is identified. In the given expression, the coefficients of x^10, x^6, and x are all zero. Thus 0x^10 = 0, 0x^6 = 0, and 0x = 0, leaving only the non-zero constant 19. A non-zero constant polynomial has degree 0. Therefore option D is correct. Option A and option B incorrectly use exponents attached to terms whose coefficients are zero; those terms are absent after simplification. Option C similarly treats the zero x-term as though it were a genuine first-degree term. This is not the zero polynomial, because the remaining constant is 19, so the degree is defined and equals 0.

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

Explanation: In \(4x^3-8x+1\), the highest power of \(x\) is \(3\), so its degree is \(3\). Since there is no \(x^2\) term, the coefficient of \(x^2\) is \(0\). In option B, the coefficient of \(x^2\) is \(1\), so it is not correct. Exam tip: The coefficient of a missing power term is always \(0\).

Correct answer: C. 6

Explanation: Expanding the expression gives \(x^4(3x^2-2)+5x^3=3x^6-2x^4+5x^3\). The highest power of \(x\) is 6, so the degree of the polynomial is 6. Choosing 4 by looking only at \(x^4\) is incorrect because it must be multiplied by each term inside the bracket. Exam tip: simplify or expand a polynomial first, then identify the highest exponent with a non-zero coefficient.

Correct answer: C. 2

Explanation: Put s = −1 into the coefficients. Then s + 1 = 0, so the x⁵ term vanishes, while s + 4 = 3, so the x² term becomes 3x². The polynomial is therefore 3x² + 8. Its highest remaining power with a non-zero coefficient is 2, hence its degree is 2. The original x⁵ term cannot determine the degree after its coefficient becomes zero.

Correct answer: C. 6

Explanation: The total degree of a polynomial is the greatest sum of the exponents of the variables in any term. Here, the total degree of 4x^3y is 3+1=4, that of 2xy^5 is 1+5=6, and the constant term 6 has degree 0. Therefore, the greatest total degree is 6. Option 4 is only the degree of the first term, not of the whole polynomial. Exam tip: Find the total degree of each term separately, then select the greatest value.

Correct answer: C. \(6x^5+7\)

Explanation: The degree of a polynomial is the greatest exponent of \(x\) with a non-zero coefficient. In \(6x^5+7\), the greatest exponent of \(x\) is \(5\), so its degree is \(5\), not \(6\). Each of the other options contains a non-zero \(x^6\) term and therefore has degree \(6\). Exam tip: identify the term with the highest power of \(x\) before deciding the degree.

Correct answer: C. (7)

Explanation: The condition v≠-3 is important because the coefficient of x^7 is v+3. If v were -3, that coefficient would become zero and the seventh-degree term would disappear. But the given condition rules out exactly that value, so the x^7 term remains present. The degree is then found by looking at the greatest exponent with a non-zero coefficient.

Since v≠-3, v+3≠0. Therefore (v+3)x^7 has a non-zero coefficient, while the other terms have powers 3 and 0. The largest exponent is consequently 7, so option C is correct. The degree is not 3 because the x^7 term has not vanished, and it is not 10 because no x^10 term occurs. This conclusion does not require knowing the exact value of v.

Correct answer: A. \(5x-7\)

Explanation: A linear polynomial has degree 1. In \(5x-7\), the highest exponent of the variable \(x\) is 1, so it is a linear polynomial. \(x^2+4x+1\) has degree 2, and \(3x^3-2\) has degree 3. \(\frac{1}{x}+5\) is not a polynomial because it contains \(x^{-1}\). Exam tip: in a polynomial, variable exponents must be non-negative integers such as 0, 1, 2, and 3.

Correct answer: B. (6)

Explanation: The direct answer is option B, 6. Substitute \(z=4\) into every coefficient. The coefficient of \(x^9\) is \(z-4=4-4=0\), so that term disappears. The coefficient of \(x^6\) is \(z-1=4-1=3\), so the expression becomes \(3x^6+3x^2-5\). The degree of a nonzero polynomial is the greatest exponent of x whose coefficient is not zero; here that exponent is 6. Thus option B is correct. Option A, 9, would be correct only if the \(x^9\) term remained, but its coefficient is zero. Option C, 2, ignores the higher nonzero term \(3x^6\). Option D, 0, would describe a nonzero constant polynomial, but the expression still contains x terms. The key lesson is that a zero coefficient removes a term, so always substitute the given value before finding the degree.

Correct answer: B. The \(x^5\) terms cancel; the resulting polynomial \(3x^2+x\) has degree 2

Explanation: When adding polynomials, combine terms with the same exponent: \(x^5+(-x^5)=0\). Thus, the resulting polynomial is \(3x^2+x\), whose highest exponent among its non-zero terms is 2. Therefore, its degree is 2. Option A ignores that the \(x^5\) terms have opposite coefficients and cancel. Exam tip: simplify the polynomial first, then identify the highest remaining exponent to find its degree.

Correct answer: D. \(n=3\)

Explanation: The degree of a polynomial is the highest power of its variable. Here, the term \(3x^5\) has power 5. Since \(n<5\), the power of \(4x^n\) is less than 5, so the polynomial still has degree 5. Thus, \(n=3\) is possible. \(n=5\) is not allowed because the condition is \(n<5\), not \(n\leq5\). Exam tip: While finding degree, always check any condition given on the exponent as well.

Correct answer: C. 5

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 5, 3, 1, and 0, and the coefficient of x^5 is 7, which is non-zero. Therefore, the degree is 5. Option 3 is only the exponent of the term -4x^3, not the highest exponent. Exam tip: A constant term has degree 0.

Correct answer: C. 4

Explanation: The coefficient of 0x^9 is 0, so it is a zero term and does not determine the degree of the polynomial. Among the remaining non-zero terms, 6x^4 has the highest exponent, 4. Therefore, the degree is 4. Choosing 9 would be incorrect because the coefficient of x^9 is zero. Exam tip: Remove terms with zero coefficients before finding the degree.

Correct answer: C. 2

Explanation: Here, 5x^6 and -5x^6 are like terms, so they cancel to 0. The expression becomes 8x^2-1. Its highest exponent is 2, so the degree of the polynomial is 2. Degree 0 applies only to a non-zero constant polynomial. Exam tip: Always simplify like terms before finding the degree.

Correct answer: C. 3

Explanation: On substituting \(b=4\), we get \(b-4=0\). Hence, the \(x^8\) term vanishes and the polynomial becomes \(3x^3+2\). The highest power of \(x\) now present is \(3\), so the degree is \(3\). Option 8 is incorrect because its coefficient becomes zero. Exam tip: Substitute any given parameter values first, then remove terms with zero coefficients before finding the degree.

Correct answer: B. \(a=-7\)

Explanation: For the polynomial to have degree 2, the coefficient of \(x^6\) must be zero. Thus, \(a+7=0\), which gives \(a=-7\). For this value, the polynomial becomes \(-4x^2+9\), whose degree is 2. If \(a=0\), the coefficient of \(x^6\) is still 7, so the degree remains 6. Exam tip: First check whether the coefficient of the highest-power term becomes zero.

Correct answer: A. (-6x^9)

Explanation: The degree is determined only after substituting the given value of the variable or parameter. A term that appears to have a high power may disappear if its coefficient becomes zero. Here, putting \(c=0\) removes the \(x^{10}\) term, so we must inspect the remaining terms rather than choosing 10 automatically. The correct option is B.

Substitute \(c=0\): \(cx^{10}=0x^{10}=0\), and \((c+4)x^6=(0+4)x^6=4x^6\). The expression becomes \(4x^6-3x+2\). Its nonzero exponents are 6, 1, and 0, so the greatest is 6. Therefore the degree is 6. Option A is wrong because the coefficient of \(x^{10}\) has become zero.

Correct answer: C. 11

Explanation: The degree of a polynomial is the greatest exponent of the variable having a non-zero coefficient. Here, the powers of x are 3, 11, and 6, while the constant term 4 has degree 0. The greatest exponent is 11, so the degree is 11. Option 6 is the degree of only one term, not of the whole polynomial. Exam tip: the order of terms does not affect the degree; identify the highest power of the variable.

Correct answer: C. 1

Explanation: Terms with coefficient 0 do not contribute to a polynomial. Thus the expression reduces to \(15x-6\). The highest power of \(x\) in this polynomial is \(1\), so its degree is \(1\). Degree \(0\) applies to a non-zero constant polynomial, not to this linear polynomial. Exam tip: first remove zero-coefficient terms, then identify the highest exponent of the variable left.

Correct answer: D. 6

Explanation: Since \(x^0=1\), the expression becomes \(3+4x^6-9x^2\). The degree of a polynomial is the greatest exponent of \(x\) in any term with a non-zero coefficient. The term \(4x^6\) has the highest exponent, 6, so the answer is 6. The \(x^0\) term is only a constant term and does not increase the degree. Exam tip: simplify terms such as \(x^0\) first, then identify the highest exponent.

Correct answer: C. 0

Explanation: 23 is a non-zero constant polynomial because it has no variable term. The degree of every non-zero constant polynomial is 0, so the correct answer is 0. A polynomial of degree 1 must have a variable with highest power 1, which is not present here. Exam tip: Any non-zero constant such as 5, -12, or 23 has degree 0 when considered as a polynomial.