Degree of a Polynomial

Correct answer: B. 5x^9-x^2+7

Explanation: The degree of a polynomial is the greatest exponent of the variable in any term with a non-zero coefficient. In option B, the term 5x^9 has exponent 9, so the polynomial 5x^9-x^2+7 has degree 9. Option A has degree 8, while the non-zero constant polynomial 30 has degree 0. Exam tip: identify the highest exponent of the variable after checking all terms.

Correct answer: C. 6

Explanation: Since \(x^0=1\), the expression becomes \(1+7x^6-4x^2\). The degree of a polynomial is the greatest exponent of \(x\) in any term, which is \(6\). Hence, the correct answer is \(6\). The number \(7\) is the coefficient of \(x^6\), not its exponent. Exam tip: Simplify terms such as \(x^0\) first, then identify the highest exponent.

Correct answer: D. 10

Explanation: The degree of a polynomial is the greatest exponent of the variable having a non-zero coefficient. Here, the exponents of the terms are 2, 10, 3, and 0. The greatest exponent is 10, so the degree of the polynomial is 10. Note that 9 is the coefficient of x^{10}, not the degree. Exam tip: To find degree, look for the highest power of the variable, not the largest coefficient.

Correct answer: B. 5

Explanation: Combining like terms gives \(6x^5+6x^5=12x^5\). Thus, the simplified polynomial is \(12x^5-3x+8\). The highest power of \(x\) is 5, so its degree is 5. Here, 12 is a coefficient, not an exponent, so 12 cannot be the degree. Exam tip: simplify the polynomial first, then identify the greatest exponent of the variable in a non-zero term.

Correct answer: C. 7

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 3, 0, and 7; the greatest is 7. Therefore, the degree is 7. The number 8 is a coefficient, not an exponent, so it is not the degree. Exam tip: Arrange terms in descending powers to identify the degree quickly.

Correct answer: C. 12

Explanation: The degree of a polynomial is the greatest exponent of its variable whose coefficient is non-zero. Since the highest power given is 12, the degree of the polynomial is 12. Degree 0 applies to a non-zero constant polynomial, while degree 1 applies to a linear polynomial. Exam tip: To find the degree, identify the greatest power of the variable.

Correct answer: B. 8

Explanation: The coefficient of \(0x^{15}\) is 0, so it is a zero term and is not considered while finding the degree. Among the remaining non-zero terms, \(2x^8\), \(-6x^4\), and \(9\), the highest exponent is 8. Therefore, the degree of the polynomial is 8. Option 15 is incorrect because the coefficient of \(x^{15}\) is zero. Exam tip: Remove terms with zero coefficients before identifying the highest power.

Correct answer: C. Constant polynomial

Explanation: The degree of a non-zero polynomial is the highest exponent of the variable whose coefficient is non-zero. If the degree is 0, no positive power of the variable is present, so the polynomial has the form c, where c is a non-zero constant; examples include 7, −3, and 1/2. Therefore it is a constant polynomial, making option C correct. A linear polynomial has degree 1, while a quadratic polynomial has degree 2. The zero polynomial is not an acceptable answer because its degree is generally undefined, not 0, since it has no non-zero coefficient and therefore no highest exponent. The word “non-zero” in the question is essential.

Correct answer: D. 0

Explanation: The key rule is that a term with a zero coefficient contributes nothing to a polynomial. Thus 0x^9 = 0, 0x^5 = 0, and 0x = 0, leaving the non-zero constant polynomial 13. A non-zero constant can be written as 13x^0, so its degree is 0. Therefore, option D is correct. The visible exponents 9, 5, and 1 do not determine the degree because their coefficients are all zero and those terms disappear. This must be distinguished from the zero polynomial: if every term vanished and the expression were exactly 0, its degree would generally be undefined. Here 13 remains, so the degree is unambiguously zero.

Correct answer: C. n = 16

Explanation: The governing concept is that the degree of a polynomial is the greatest exponent attached to a term with a nonzero coefficient. In 3x^n + 4x^13 − 6, the coefficient of x^n is 3, so the x^n term definitely remains. The condition n > 13 means its exponent is strictly greater than 13, so the degree is indeed n. We therefore check which listed value is strictly greater than 13. Only 16 satisfies that condition. Values 11 and 9 are less than 13, and 13 is equal to 13 rather than greater than 13. Thus n = 16 is the only possible value, so option C is correct. No cancellation issue arises because the coefficient 3 is nonzero.

Correct answer: A. 15

Explanation: The governing concept is that the degree of a nonzero polynomial is the greatest exponent having a nonzero coefficient. Here the coefficient of x^15 is b−6. The condition b ≠ 6 guarantees that b−6 is not zero, so the x^15 term remains present. Since 15 is greater than 6 and 0, this term has the highest exponent, and the polynomial has degree 15. Therefore option A is correct. If b were equal to 6, the x^15 term would disappear and the degree would become 6 because 9x^6 remains. That possible special case is excluded by the question. Options B, C, and D therefore confuse a lower power with the actual highest surviving power.

Correct answer: A. 7x^11 − 5x^9 + 1

Explanation: To satisfy both conditions, inspect the highest exponent and check whether an x^10 term is present. In option A, the highest exponent is 11 and its coefficient is 7, so the degree is 11. There is no x^10 term; an absent term has coefficient 0. Thus option A meets both requirements. Option B has degree 11 but contains x^10 with coefficient 1, so it fails the second condition. Option C has no x^11 term and therefore degree 10, while option D has the term x^12 and therefore degree 12. The correct answer is consequently option A, based on both degree and coefficient conditions rather than on only one of them.

Correct answer: D. n=17

Explanation: For a polynomial, the degree is the greatest exponent of x having a nonzero coefficient. In 4x^n+5x^14−7, the coefficient of x^n is 4, which is nonzero, and the coefficient of x^14 is 5, also nonzero. The condition n>14 means that x^n has a higher power than x^14, so the degree is indeed n. We only need to choose an option satisfying n>14. Values 12 and 10 are less than 14, so they violate the stated condition. The value 14 does not satisfy the strict inequality n>14. The value 17 is greater than 14 and therefore makes x^17 the highest-degree term, confirming that the degree is 17. Thus option D is the only possible answer.

Correct answer: B. 12

Explanation: The degree must be determined after substituting the given value of the parameter. With e=5, the coefficient of x^14 becomes e−5=5−5=0, so the x^14 term disappears. The coefficient of x^12 becomes e+3=5+3=8, which is nonzero, so the x^12 term remains. The expression therefore becomes 8x^12+9x^5−4. Since 8 is nonzero and 12 is the largest exponent present, the degree is 12. Option A incorrectly keeps the vanished x^14 term. Option C identifies the degree of a lower term rather than the highest remaining term, and option D confuses the constant term with the polynomial's degree. Thus option B is correct.

Correct answer: B. 17

Explanation: A zero coefficient makes its entire term equal to zero, so 0x^19 is not a genuine term of the polynomial and cannot determine its degree. Because t≠0, the coefficient of x^17 is nonzero, so tx^17 remains present. The other nonconstant term is 8x^10, whose coefficient is also nonzero, while −7 is a constant term of degree 0. Among the actual nonzero terms, the exponents are 17 and 10, and the larger exponent is 17. Therefore the polynomial has degree 17. Option A incorrectly treats the written exponent 19 as active despite its zero coefficient. Option C overlooks the x^17 term, and option D refers only to the constant term. Hence option B is correct.

Correct answer: B. Because its degree is 2

Explanation: The governing concept is the degree of a polynomial, which is the greatest exponent of the variable having a nonzero coefficient. Multiplying the two factors gives (x + 3)(2x − 1) = 2x² − x + 6x − 3 = 2x² + 5x − 3. The highest power of x is 2, and the coefficient of x² is 2, not zero. Therefore, the product has degree 2 and is a quadratic polynomial, not a linear polynomial. A linear polynomial must have degree exactly 1, so option B is correct. Option A would describe a linear expression, while option C is wrong because the result still contains x. Option D is also wrong because the expression is not identically zero; the x² term does not cancel.

Correct answer: B. Because its degree is 2

Explanation: The relevant concept is determining the degree after multiplying polynomials. Expand the product carefully: (2x + 3)(x − 4) = 2x·x + 2x(−4) + 3x + 3(−4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12. The largest exponent of x is 2, and its coefficient is 2, which is nonzero. Consequently, the product has degree 2 and is a quadratic polynomial. A linear polynomial has degree exactly 1, so the product cannot be linear. Option B is therefore correct. Option A confuses the degree of each individual factor with the degree of their product. Options C and D are not possible because the resulting expression contains nonzero x and x² terms and is neither constant nor the zero polynomial.

Correct answer: C. 3

Explanation: The degree of a polynomial is the greatest exponent of the variable in any term with a non-zero coefficient. The terms in the given polynomial have degrees 3, 2, 1, and 0. The greatest degree is 3, so the correct answer is 3. Option 5 is the constant term, not the degree. Exam tip: To find a polynomial’s degree, identify the term with the highest power of the variable.