Degree of a Polynomial

Correct answer: C. It is not defined

Explanation: The zero polynomial has no non-zero term, so there is no highest exponent to identify. Hence, its degree is not defined. Exam tip: do not confuse it with a non-zero constant polynomial, whose degree is 0.

Correct answer: A. \(0\)

Explanation: The zero polynomial \(0\) has no non-zero term, so there is no greatest exponent to identify. Therefore, its degree is undefined. A non-zero constant such as \(7\) has degree 0. Exam tip: do not confuse the zero polynomial with a non-zero constant polynomial.

Correct answer: B. 5

Explanation: While subtracting the second bracket, the sign of every term changes: p(x)=x^8-6x^5+3-x^8+2x^5=-4x^5+3. The x^8 terms cancel, and the highest power of x in the remaining polynomial is 5. Hence, its degree is 5. Option 8 is a close distractor, but the coefficient of x^8 becomes 0 after simplification. Exam tip: Always simplify a polynomial completely before finding its degree.

Correct answer: B. 8

Explanation: On expanding, \(x^8(x^2+1)-x^2(x^8-7)=x^{10}+x^8-x^{10}+7x^2=x^8+7x^2\). The \(x^{10}\) terms cancel each other. The highest power of \(x\) in the remaining polynomial is \(8\), so its degree is \(8\). Option \(10\) would result from overlooking the cancellation of like terms. Exam tip: Always simplify a polynomial before finding its degree.

Correct answer: D. Not defined

Explanation: On simplification, p(x)=0 because every term has coefficient 0. This is the zero polynomial. It has no non-zero term with a highest power, so its degree is not defined. Option 0 is the degree of a non-zero constant polynomial, such as p(x)=5, not of the zero polynomial. Exam tip: Simplify a polynomial and remove zero-coefficient terms before finding its degree.

Correct answer: A. The degree of the product equals the sum of the degrees of the two polynomials.

Explanation: If the leading terms are \(ax^m\) and \(bx^n\), their product has leading term \(abx^{m+n}\). Hence its degree is \(m+n\). Exam tip: add degrees while multiplying non-zero polynomials; do not subtract them.

Correct answer: B. \(x^7y^3+2x\)

Explanation: The total degree of a polynomial is the greatest sum of the exponents of variables in any one term. In \(x^7y^3+2x\), the term \(x^7y^3\) has total degree \(7+3=10\), while \(2x\) has degree 1. Hence, the polynomial has total degree 10. In option A, the greatest total degree is \(5+4=9\), so it is not correct. Exam tip: add the exponents in each term and select the largest sum.

Correct answer: B. (16)

Explanation: The degree of a polynomial is the greatest exponent of the variable whose coefficient is not zero. A term with coefficient zero contributes nothing, even if the exponent written beside it is larger. The condition \(t\neq0\) is important because it guarantees that the term \(tx^{16}\) is genuinely present.

In the expression, \(0x^{18}=0\), so the apparent degree-18 term disappears. The remaining nonzero terms are \(tx^{16}\), \(7x^9\), and \(-6\). Their exponents are 16, 9, and 0, and the greatest is 16. Therefore the correct choice is B, (16). Choice A incorrectly counts the zero term.

Correct answer: C. 0

Explanation: On subtraction, the like terms cancel:
\((x^8+x^5+1)-(x^8+x^5)=1\). The resulting polynomial, 1, is a non-zero constant polynomial, so its degree is 0. Although 8 and 5 occur in the original expression, those terms do not remain after simplification. Exam tip: Always simplify a polynomial before finding its degree; the zero polynomial has undefined degree, but the degree of 1 is 0.

Correct answer: B. (s=-11)

Explanation: For degree (0), both (x^{13}) and (x^9) terms must vanish. At (s=-11), both coefficients become (0).

Correct answer: D. 0

Explanation: On subtracting the polynomials, the like terms cancel: \(A(x)-B(x)=(x^{10}+6x^5+13)-(x^{10}+6x^5)=13\). This is a non-zero constant polynomial, so its degree is \(0\). Although the original polynomials have degree \(10\), their highest-degree terms cancel on subtraction. Exam tip: always find the simplified resulting polynomial before identifying its degree.

Correct answer: B. 6

Explanation: \(A(x)+B(x)=(x^9+8)+(-x^9+10x^6)=10x^6+8\). The \(x^9\) and \(-x^9\) terms cancel each other. The highest exponent in the remaining polynomial is 6, so its degree is 6. Option 9 is incorrect because no \(x^9\) term remains after addition. Exam tip: Always simplify the polynomial first, then identify its highest exponent.

Correct answer: C. The student is incorrect; after combining like terms, the \(x^4\) and \(x^3\) terms cancel, so the degree is 2.

Explanation: The coefficient of \(x^4\) is \(3+5-8=0\), and that of \(x^3\) is \(-2+2=0\). Thus \(R(x)=x^2\), whose degree is 2. In exams, simplify before finding degree.

Correct answer: B. 14

Explanation: Substituting the given values gives the polynomial \(5x^9+7x^{14}+1\). The degree of a polynomial is the greatest exponent of \(x\) in any term, so its degree is \(14\). \(23\) is the sum of the exponents, but exponents are not added to find a polynomial’s degree. Exam tip: first write the polynomial in simplified form and then identify the highest power of \(x\).

Correct answer: D. \(x^8-x^8+11x^5\)

Explanation: In option D, \(x^8-x^8=0\), so the expression simplifies to \(11x^5\). Its highest exponent is 5, so its degree is 5, not 8. In option B, \(0x^9\) has no effect, and the highest remaining power is \(x^8\). Exam tip: simplify like terms before finding the degree of a polynomial.

Correct answer: C. \(0\)

Explanation: First expand: \(x^6(x^3+x)=x^9+x^7\). Hence, \(p(x)=x^9+x^7-x^9-x^7+41=41\). A non-zero constant polynomial such as \(41\) has degree \(0\), so \(0\) is correct. Choosing \(9\) or \(7\) ignores the cancellation of like terms. Exam tip: expand brackets and combine like terms before finding a polynomial’s degree.

Correct answer: C. 8

Explanation: This polynomial is a product of two factors. The degree of x^6 is 6, and the degree of (x-5)^2 is 2. Degrees add when polynomials are multiplied, so the total degree is 6+2=8. Option 12 is incorrect because the exponents are not multiplied. Exam tip: For a polynomial written as a product, add the degrees of its factors.

Correct answer: A. \(r=10\)

Explanation: For the polynomial to have degree 0, it must reduce to a constant, namely 22. Therefore, the coefficients of both \(x^{11}\) and \(x^4\) must be zero. Since each coefficient is \(r-10\), we get \(r-10=0\), so \(r=10\). If \(r=-10\), the coefficient becomes \(-20\), and the polynomial still has degree 11. Exam tip: In parameter-based polynomials, first set the coefficient of the highest power to zero when checking whether the degree can decrease.

Correct answer: C. 15

Explanation: For non-zero polynomials, the degree of a product equals the sum of their degrees. Thus, the degree of ( p(x)q(x) ) is 8 + 7 = 15. Option 8 is only the degree of p(x), not of the product. Exam tip: Add degrees when multiplying polynomials; for addition, the higher degree generally determines the result.

Correct answer: B. 13

Explanation: For a term in two variables, the total degree is the sum of the exponents of its variables. Here, the total degrees of x^7y^2, x^5y^8, and xy are 7+2=9, 5+8=13, and 1+1=2 respectively. The greatest of these is 13, so the total degree of the polynomial is 13. Option A is only the total degree of the first term. Exam tip: Find the total degree of every term and choose the greatest one.

Correct answer: C. 5

Explanation: In the given expression, \(6x^{12}-6x^{12}=0\) and \(7x^8-7x^8=0\). Thus, \(F(x)=9x^5-4\). The highest exponent with a non-zero coefficient is \(5\), so the degree of the polynomial is \(5\). Option 0 would apply only to a non-zero constant polynomial. Exam tip: simplify like terms first, then identify the highest remaining exponent.

Correct answer: C. (6)

Explanation: ((x^7+4x^3)^2=x^{14}+8x^{10}+16x^6). After higher terms cancel, (16x^6+6x^2) remains, whose degree is (6).

Correct answer: A. (a=-6)

Explanation: For degree (4), both (x^{13}) and (x^9) terms must vanish. At (a=-6), both higher-term coefficients become (0).

Correct answer: B. (8)

Explanation: The direct answer is option B, degree 8. Distribute \(x^7\): \(x^7(x^4-3)=x^{11}-3x^7\). Substitution gives \(x^{11}-3x^7-x^{11}+5x^8-6\). The two degree-11 terms cancel: \(x^{11}-x^{11}=0\). The simplified polynomial is \(5x^8-3x^7-6\), whose highest surviving exponent is 8. Therefore option B is correct. Option A, 11, is wrong because both degree-11 terms disappear. Option B, 8, is correct because \(5x^8\) remains and its coefficient is non-zero. Option C, 7, overlooks the higher surviving term \(5x^8\). Option D, 4, is not present in the simplified expression. The common mistake is to look at the largest exponent before combining like terms. First expand, then cancel, then identify the largest remaining exponent.

Correct answer: A. \(\deg(pq)=\deg p+\deg q\)

Explanation: When the leading terms of two non-zero polynomials are multiplied, their exponents add and the resulting leading coefficient remains non-zero. Hence \(\deg(pq)=\deg p+\deg q\). Exam tip: this rule requires both polynomials to be non-zero.