Degree of a Polynomial

Correct answer: C. (6)

Explanation: ((x^6+2x^3)^2=x^{12}+4x^9+4x^6). After higher terms cancel, (4x^6+7x^5) remains, whose degree is (6).

Correct answer: A. (w=10)

Explanation: For degree (2), both (x^{14}) and (x^8) terms must vanish. At (w=10), both coefficients become (0).

Correct answer: C. (8)

Explanation: The product creates an (x^{12}) term and the outside (-x^{12}) cancels it. The remaining highest power is (8).

Correct answer: A. (a=5)

Explanation: For degree (2), both (x^{12}) and (x^8) terms must vanish. At (a=5), both higher-term coefficients become (0).

Correct answer: B. (7)

Explanation: The direct answer is option B, degree 7. First distribute \(x^6\): \(x^6(x^4-2)=x^{10}-2x^6\). The whole polynomial becomes \(x^{10}-2x^6-x^{10}+3x^7-9\). The like terms \(x^{10}-x^{10}\) cancel to zero, leaving \(3x^7-2x^6-9\). The highest exponent with a non-zero coefficient is 7, so the degree is 7. Option A, 10, is wrong because the two degree-10 terms cancel. Option B, 7, is correct. Option C, 6, is wrong because the non-zero term \(3x^7\) has a higher exponent than \(-2x^6\). Option D, 4, is unrelated; no degree-4 term remains. Always simplify before finding degree, because leading terms can cancel. Memory cue: degree means the largest exponent that survives with a non-zero coefficient.

Correct answer: C. (3)

Explanation: Direct answer: Option C, degree 3. Substitute k=-2 carefully. Then \\(k+2=0\\), so \\( (k+2)^2=0\\), removing the x^11 term. Also \\(k^2-4=(-2)^2-4=4-4=0\\), removing the x^6 term. The polynomial becomes \\(q(x)=-5x^3+1\\). Its highest power with a nonzero coefficient is x^3, so its degree is 3. Option A, 11, ignores that the x^11 coefficient becomes zero. Option B, 6, ignores that the x^6 coefficient also becomes zero. Option C correctly identifies the surviving highest power. Option D, 0, would be correct only if the x^3 term also disappeared, but its coefficient -5 is nonzero. Always substitute the given parameter into every coefficient before reading the degree. Memory cue: zero coefficient means the term is absent.

Correct answer: C. She ignored the exponent 4 in \(-3x^4\)

Explanation: The degree of a polynomial is the greatest exponent, not the exponent of the last written term. Here the exponents are 0, 4 and 2, so the degree is 4. The term \(-3x^4\) decides it. Exam tip: list exponents and select the largest.

Correct answer: B. 5

Explanation: Combining like terms gives \(6x^9-11x^9+5x^9=(6-11+5)x^9=0\). Thus, \(f(x)=4x^5-13\). The highest exponent is 5, so the degree of the polynomial is 5. Option 9 is incorrect because all the \(x^9\) terms cancel completely. Exam tip: Before finding the degree, simplify like terms and check whether the leading coefficient becomes zero.

Correct answer: B. 11

Explanation: The total degree of a polynomial is the greatest sum of the exponents of the variables in any one term. The term degrees here are \(6+4=10\), \(2+9=11\), \(1+3=4\), and \(0\) for the constant term \(-7\). Hence, the highest total degree is \(11\). Although \(10\) is the degree of the first term \(8x^6y^4\), it is not the greatest. Exam tip: add the exponents within each term, then select the largest sum.

Correct answer: B. \(9\)

Explanation: To find the degree with respect to \(x\), treat every power of \(y\) as part of the coefficient. The powers of \(x\) in the expression are \(7\) and \(9\); in \(6y^8-4\), the power of \(x\) is \(0\). Since the greatest power is \(9\), the degree of the polynomial with respect to \(x\) is \(9\). Option \(7\) is only the power of \(x\) in the first term, \(9x^7y^5\). Exam tip: for degree with respect to one variable, compare powers of that variable only.

Correct answer: C. 8

Explanation: To find the degree with respect to y, consider only the powers of y and treat x as a constant. The powers of y in the given terms are 5, 3, 8, and 0 respectively. The greatest power is 8, occurring in 6y^8. Hence, the degree with respect to y is 8. Although 9x^7y^5 has total degree 12, that is not its degree in y. Exam tip: for degree in one variable, compare exponents of that variable only.

Correct answer: D. \(m=8\)

Explanation: The degree of a polynomial is the highest power of its variable. The term \(2x^{10}\) is already present. Since \(m<10\), taking \(m=8\) keeps \(10\) as the highest exponent, so the polynomial has degree \(10\). Although \(m=10\) would also leave the degree as 10, it does not satisfy the condition \(m<10\). Exam tip: While finding a degree, always check any condition given on the exponent as well.

Correct answer: C. 18

Explanation: The polynomial \((x^6-4)\) has degree 6. When a non-zero polynomial is raised to a power, its degree is multiplied by that power: \(6\times3=18\). On expansion, the highest-degree term is \((x^6)^3=x^{18}\), so the degree is 18. Choosing 12 may result from an incorrect operation on exponents; here the degree must be multiplied by 3. Exam tip: \(\deg((P(x))^n)=n\deg(P(x))\) for a non-zero polynomial \(P(x)\).

Correct answer: B. \(\deg(PQ)=10\)

Explanation: For non-zero polynomials, the degree of a product equals the sum of their degrees. Hence \(\deg(PQ)=4+6=10\). In addition or subtraction, leading terms may cancel, so the degree is not fixed. Exam tip: add degrees only when polynomials are multiplied.

Correct answer: B. It is the greatest exponent of the variable with a non-zero coefficient.

Explanation: The degree of a non-zero polynomial is the highest exponent of its variable whose coefficient is non-zero. The number of terms is not its degree. Exam tip: ignore every term with coefficient 0 first.

Correct answer: B. The student is incorrect; the degree is 3.

Explanation: Since \(4x^5-4x^5=0\), the simplified polynomial is \(6x^3-x+9\). Its highest non-zero exponent is 3, so its degree is 3. Exam tip: always simplify like terms before finding degree.

Correct answer: B. 8

Explanation: First expand: \(6x^8(x^2-1)=6x^{10}-6x^8\). Therefore, the full expression becomes \(6x^{10}-6x^8-6x^{10}+5x^7=-6x^8+5x^7\). The \(x^{10}\) terms cancel, and the highest power in the remaining polynomial is 8, so its degree is 8. Option 10 is incorrect because the coefficient of \(x^{10}\) becomes zero after simplification. Exam tip: Always expand brackets and combine like terms before finding a polynomial’s degree.

Correct answer: D. (0)

Explanation: Terms with zero coefficients disappear and (53) remains. It is a non-zero constant polynomial with degree (0).

Correct answer: C. Not defined

Explanation: Here, \(Z(x)=0\) is the zero polynomial for every value of \(x\). It has no non-zero term, so there is no highest power to determine its degree. Therefore, its degree is not defined. Degree \(0\) applies to a non-zero constant polynomial, not to the zero polynomial. Exam tip: Before finding degree, check whether the polynomial is identically zero.

Correct answer: D. 0

Explanation: First simplify the expression: \(x^6(x^2-7)=x^8-7x^6\). Hence, \(p(x)=x^8-7x^6-x^8+7x^6+31=31\). This is a non-zero constant polynomial, so its degree is \(0\). Choosing 8 would ignore the cancellation of like terms. Exam tip: always simplify a polynomial completely before finding its degree.

Correct answer: B. (9)

Explanation: When multiplying polynomials, every term in the first factor must be multiplied by every term in the second factor. Like powers can then be combined, and any term that is subtracted may cancel an identical term. The degree is determined only after this simplification, because the apparent highest power may disappear.

Expand the product: \(x^6(x^7-x^3)=x^{13}-x^9\), and \(3x^2(x^7-x^3)=3x^9-3x^5\). Adding gives \(x^{13}+2x^9-3x^5\). Subtracting \(x^{13}\) cancels the degree-13 term, leaving \(2x^9-3x^5\). The highest remaining exponent is 9, so option B is correct. Choosing 13 would overlook the cancellation.

Correct answer: B. (c=-7)

Explanation: The direct answer is option B, \(c=-7\). For degree 0, the terms with \(x^9\) and \(x^6\) must both disappear, because any surviving positive power would give degree at least 6. The coefficients are \(c+7\) and \(c^2-49\). Setting \(c=-7\) gives \(c+7=0\) and \(c^2-49=49-49=0\). The expression becomes the non-zero constant \(43\), so its degree is 0. Option A, \(c=7\), makes the second coefficient zero but leaves the first as 14, so degree 9 remains. Option B makes both coefficients zero. Option C leaves both variable terms present. Option D also leaves a non-zero \(x^9\) coefficient. Memory cue: when expressions contain \(c+a\) and \(c^2-a^2\), try \(c=-a\) to remove both terms.

Correct answer: B. (d=8)

Explanation: Direct answer: Option B, \\(d=8\\). To make the degree 5, the x^8 term must vanish and the x^5 term must remain. The x^8 coefficient is d-8, so d-8=0 gives d=8. At this value, the x^5 coefficient is d+2=10, which is nonzero. The polynomial becomes \\(10x^5+17\\), so its degree is 5. Option A, d=-2, removes the x^5 term but leaves coefficient -10 on x^8, giving degree 8. Option B satisfies both required conditions. Option C, d=0, leaves a nonzero x^8 coefficient -8, so degree 8. Option D, d=5, leaves coefficient -3 on x^8, again giving degree 8. The important check is two-part: cancel every power higher than the requested degree, then verify the requested power survives.

Correct answer: B. (9)

Explanation: The phrase “with respect to z” tells us to inspect only the exponents of z. Any powers of x are included in the coefficients for this purpose, so they do not determine the requested degree. A term without z has z-degree zero, while a term containing z has the exponent displayed on z. The largest such exponent gives the degree of the polynomial in z.

The z-powers in the expression are 9 in the first term and 4 in the second term. The remaining terms contain no z, so they have z-degree zero. Since 9 is the largest exponent and the coefficient of z^9 is nonzero, the degree is 9. Therefore, option B is correct; choosing 5 would confuse the power of x with the power of z.

Correct answer: C. \(6\)

Explanation: \((x^6+x^3)^2=x^{12}+2x^9+x^6\). Therefore, \(h(x)=x^{12}+2x^9+x^6-x^{12}-2x^9=x^6\). The highest exponent in the simplified polynomial is \(6\), so its degree is \(6\). The terms with exponents \(12\) and \(9\) cancel, so they do not determine the degree. Exam tip: simplify the expression and cancel like terms before finding the degree.