Degree of a Polynomial

Correct answer: D. 10

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. Here, the powers of x are 4, 6, 8, and 10. The greatest power is 10, so the degree of the polynomial is 10. Option 8 is a close distractor, but the term x^{10} is present, so the degree cannot be 8. Exam tip: Before finding the degree, check the exponents of the variable in every term.

Correct answer: A. \(5x^3-2x+7\)

Explanation: A cubic polynomial has highest exponent 3. In \(5x^3-2x+7\), the greatest power is 3, so it is cubic. \(x^4+x\) has degree 4. Exam tip: check the greatest exponent.

Correct answer: B. 12

Explanation: The degree of a polynomial in one variable is the highest exponent of the variable with a non-zero coefficient. Here, the powers of x are 12, 9, 4, and 0. The greatest power is 12, so the degree is 12. The number 2 is a coefficient, not the degree. Exam tip: Treat a non-zero constant term as having degree 0.

Correct answer: C. 11

Explanation: The degree of a polynomial is the greatest exponent of the variable among terms with non-zero coefficients. Since n>6, the exponent in 5x^n is greater than 6, so it is the highest exponent. Hence, the degree of the polynomial is n. Given that the degree is 11, n=11. Option 6 is incorrect because x^6 is only the term with the next lower exponent. Exam tip: identify the highest exponent first and check that its coefficient is non-zero.

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

Explanation: The degree depends on the highest power that has a non-zero coefficient. In x^6+ax^2+5, the coefficient of x^6 is fixed at 1. Since this coefficient cannot be zero, the sixth-degree term remains present for every real value of a. The coefficient a belongs only to the lower-power x^2 term and cannot change the leading degree.

If a=0, the expression is x^6+5, which still has degree 6. If a is non-zero, an x^2 term is added, but it is still lower than x^6. Thus all real values of a produce a polynomial of degree 6. Option A is correct. There is no need for a to equal 6, and the condition a≠0 is too strong because a=0 also satisfies the requirement.

Correct answer: D. 0

Explanation: The governing rule is to simplify a polynomial by removing every term whose coefficient is zero before identifying the highest power. Here 0x^12, 0x^7, and 0x are all zero terms. After they disappear, the expression becomes simply 37. Since 37 is a non-zero constant, its polynomial degree is 0, making option D correct. Option A and option B incorrectly retain the exponents of zero terms, even though those terms have no effect on the expression. Option C similarly treats 0x as a genuine linear term, but its coefficient is zero, so it vanishes. The result is not the zero polynomial because 37 remains; this is why the degree is defined as 0 rather than being treated as undefined.

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

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

Correct answer: C. 7

Explanation: Expanding the expression gives \(x^5(2x^2-3)+4x^4=2x^7-3x^5+4x^4\). The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. Here, the greatest exponent is \(7\), so the degree is \(7\). Option \(5\) is only the exponent in one term, not the degree of the whole polynomial. Exam tip: Expand the brackets first, then identify the highest power.

Correct answer: C. 3

Explanation: Substitute s = 2. The coefficient of x⁶ becomes s − 2 = 0, so the sixth-degree term disappears. The coefficient of x³ becomes s + 5 = 7, which is non-zero. The expression reduces to 7x³ + 9, whose highest exponent is 3. Therefore its degree is 3; the vanished x⁶ term and the constant 9 do not change this result.

Correct answer: C. 7

Explanation: The total degree of a polynomial is the greatest total degree among its terms. For 3x^4y, the total degree is 4+1=5; for 5xy^6, it is 1+6=7; and the constant term 2 has degree 0. Hence, the greatest total degree is 7. Option 5 is only the degree of the first term, not of the whole polynomial. Exam tip: Add the exponents of all variables in each term and select the largest sum.

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

Explanation: The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. In \(7x^6+5\), the highest power of \(x\) is 6, so its degree is 6, not 7. In contrast, \(x^7+2\) has the term \(x^7\), so its degree is 7. Exam tip: Simplify the polynomial first, then identify the highest exponent of the variable.

Correct answer: C. 8

Explanation: Since \(v\neq5\), \(v-5\neq0\). Hence, the coefficient of \(x^8\) is non-zero, so \((v-5)x^8\) is the highest-degree term of the polynomial. Therefore, its degree is \(8\). The term \(x^4\) has degree \(4\), so it cannot determine the degree here. Exam tip: In a polynomial containing a parameter, first check whether the coefficient of the highest-power term becomes zero.

Correct answer: B. The degree of a polynomial is determined by the greatest exponent of the variable, not by the order of its terms.

Explanation: The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient; it does not depend on the order in which terms are written. The term \(-3x^4\) has exponent 4, so the polynomial has degree 4. Although \(x^2\) is the last term, it does not determine the degree. Exam tip: identify the term with the highest power before stating the degree.

Correct answer: B. (6)

Explanation: When a parameter is assigned a value, substitute it before finding the degree. The highest-power term counts only if its resulting coefficient is nonzero. This is important because a coefficient such as \(c-2\) can become zero and make the corresponding term disappear. After substitution here, the sixth-power term remains, so option B is correct.

Put \(c=0\) into the expression. The first term becomes \(0x^{10}=0\), while the second coefficient becomes \(c+4=4\). Thus the polynomial reduces to \(4x^6-3x+2\). The greatest exponent of \(x\) in a nonzero term is 6, so its degree is 6. The apparent power 10 cannot be used because its entire term has vanished.

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

Explanation: In \(4x^3-5x+2\), the highest power of \(x\) is 3, and all powers of \(x\) are non-negative integers. Hence, it is a polynomial of degree three. Option B has \(\frac{2}{x}=2x^{-1}\), so it is not a polynomial. Option C is a polynomial but has degree 4, while option D contains \(\sqrt{x}=x^{1/2}\). Exam tip: first verify that every exponent is a non-negative integer, then find the highest exponent.

Correct answer: B. d = −2

Explanation: The governing concept is that the degree of a nonzero polynomial is the greatest exponent whose coefficient is nonzero. For the given polynomial to have degree 3, the x^5 term must disappear; therefore its coefficient must satisfy d+2=0, giving d=−2. We must also check that the x^3 term does not disappear at this value. Its coefficient becomes d−4=−2−4=−6, which is nonzero. Hence the polynomial becomes −6x^3+9, whose highest nonzero power is x^3. Option B is correct. If d=4, the x^3 term vanishes but the x^5 term remains. For d=0 or d=5, the x^5 coefficient is also nonzero, so those choices leave the degree equal to 5.

Correct answer: C. 13

Explanation: The degree of a non-zero polynomial in one variable is the greatest exponent of the variable whose coefficient is non-zero. In this expression, the coefficient of x^13 is b−4. The condition b≠4 guarantees that b−4 is not zero, so the x^13 term remains present. The other variable term, 6x^7, has degree 7, and the constant term −2 has degree 0. Since 13 is the greatest exponent attached to a term with a non-zero coefficient, the polynomial has degree 13. Thus option C is correct. Option A refers only to the lower-power term, while 6 and 4 are neither the highest surviving exponent nor the degree of the complete polynomial.

Correct answer: B. 9

Explanation: The governing concept is that the degree of a nonzero polynomial is the greatest exponent whose coefficient is nonzero. Substitute e = −1 before deciding the degree. The coefficient of x^11 is e + 1 = −1 + 1 = 0, so the complete x^11 term vanishes and cannot determine the degree. The coefficient of x^9 is e − 2 = −1 − 2 = −3, which is nonzero, so that term remains. The terms 5x^3 and −6 also remain, but their degrees are only 3 and 0. Thus the simplified polynomial has highest surviving exponent 9, so option B is correct. Option A incorrectly counts the cancelled term; options C and D refer only to lower-degree surviving terms.

Correct answer: C. n = 15

Explanation: The degree of a polynomial is the greatest exponent of the variable whose coefficient is non-zero. Here the coefficient of x^n is 2, so the x^n term definitely exists. The condition n>12 means that this exponent is larger than the exponent 12 in 9x^12. Therefore the degree is indeed n, provided n is greater than 12. Among the given choices, only n=15 satisfies that strict inequality. The values n=10 and n=8 are smaller than 12, and n=12 is equal to 12 rather than greater than 12. Thus option C is the only possible answer. The non-zero coefficient 2 is important because it prevents the leading term from disappearing.

Correct answer: B. 10

Explanation: The governing rule is that the degree of a nonzero polynomial is the greatest exponent with a nonzero coefficient. Substitute e = 3 into the coefficients. For x^12, e − 3 = 3 − 3 = 0, so the term 0x^12 disappears completely. For x^10, e + 2 = 3 + 2 = 5, so the term 5x^10 remains. The expression therefore becomes 5x^10 + 6x^4 − 7. Its surviving exponents are 10, 4, and 0, and the greatest is 10. Hence option B is correct. Option A mistakenly retains the cancelled x^12 term. Option C is only the degree of the x^4 term, while option D represents the constant term rather than the polynomial’s degree.

Correct answer: A. 13

Explanation: For a polynomial in several variables, the total degree of a term is the sum of the exponents of all variables in that term. The first term x^6y^3z^4 has total degree 6 + 3 + 4 = 13. The second term 4x^2y^2z^9 has total degree 2 + 2 + 9 = 13; its numerical coefficient 4 does not affect degree. The constant term −12 has degree 0. The degree of the polynomial is the largest term degree, so it is 13. Therefore option A is correct. Adding the exponents across a term is essential; taking only the largest exponent would incorrectly suggest 9 or 6, and ignoring the first term would miss the maximum.

Correct answer: B. 11

Explanation: Substitute e = −4 before determining the degree. The coefficient of x^13 becomes e+4 = −4+4 = 0, so the x^13 term vanishes completely. The coefficient of x^11 becomes e−1 = −4−1 = −5, which is nonzero, so the x^11 term remains. The lower term 8x^4 also remains, but its exponent is smaller than 11. Therefore the highest exponent with a nonzero coefficient is 11, and option B is correct. Option A overlooks the zero coefficient of x^13; option C ignores the surviving x^11 term; option D treats the constant term as the degree even though higher nonzero powers remain.

Correct answer: B. u=-12

Explanation: The governing concept is that the degree of a nonzero polynomial is the greatest exponent whose coefficient is nonzero. For the expression to have degree 0, both variable terms must disappear, while the constant 11 must remain. The coefficient of x^14 is u^2−144=(u−12)(u+12), and the coefficient of x^7 is u+12. The second coefficient becomes zero only when u=−12. Substitution into the first coefficient gives (−12)^2−144=144−144=0 as well. Thus the expression reduces to 11, a nonzero constant polynomial, whose degree is 0. The choice u=12 removes the x^14 term but leaves 24x^7, and the other values do not remove both terms. Therefore option B is correct.

Correct answer: C. 2

Explanation: The governing rule is that the degree is determined only after all given parameter values have been substituted and like terms or zero terms have been identified. Put k=3 into the expression. The coefficient of x^12 is (3−3)^2=0, so that term vanishes. The coefficient of x^7 is 3^2−9=9−9=0, so this term also vanishes. The remaining polynomial is q(x)=6x^2−5. Since the coefficient 6 of x^2 is nonzero, the highest power present is 2, and therefore the degree is 2. Options 12 and 7 incorrectly retain terms whose coefficients become zero; option 0 confuses the constant term with the degree of the whole nonconstant polynomial. Hence option C is correct.

Correct answer: D. 8x^{12}-6x^{10}+1

Explanation: The degree is the highest exponent with a nonzero coefficient, while the coefficient of a missing power is understood to be zero. In option A, the degree is 12, but x^11 is present with coefficient 1, so it fails the second condition. In option B, x^11 has coefficient 8 and the degree is only 11. In option C, the term x^13 is present, so the degree is 13 rather than 12. In option D, the leading term 8x^12 has a nonzero coefficient, so the degree is 12, and there is no x^11 term at all; consequently its coefficient is 0. The term −6x^10 does not affect either required condition. Therefore option D satisfies both requirements.