Degree of a Polynomial

Correct answer: A. (s=2)

The direct answer is A, \(s=2\). For the polynomial to have degree 0, the coefficients of both positive-power terms, \(x^9\) and \(x^7\), must be zero. Set the first coefficient to zero: \(s^2-4=0\), so \(s^2=4\) and \(s=2\) or \(s=-2\). Now use the second coefficient, \(s-2=0\), which gives \(s=2\). At \(s=2\), the polynomial becomes the nonzero constant 4, so its degree is 0. Option A is correct. Option B, \(s=-2\), makes the first coefficient zero but gives \(s-2=-4\), leaving an \(x^7\) term. Option C, \(s=0\), leaves both variable terms nonzero. Option D, \(s=4\), also leaves both coefficients nonzero. The important method is to impose all required coefficient conditions, not stop after solving only one equation.

Correct answer: D. 0

On subtraction, the like terms cancel: \(A(x)-B(x)=(x^7+3x^2+5)-(x^7+3x^2)=5\). This is a non-zero constant polynomial, so its degree is \(0\). Although the original polynomials have degree \(7\), their highest-degree terms cancel after subtraction. Exam tip: always simplify the polynomial first, then identify its highest power of the variable.

Correct answer: C. 3

\(A(x)+B(x)=(x^6+2)+(-x^6+4x^3)=4x^3+2\). The \(x^6\) terms cancel each other. The highest exponent in the resulting polynomial is 3, so its degree is 3. Option 6 is incorrect because no \(x^6\) term remains after addition. Exam tip: Find the degree only after simplifying the sum and identifying the highest power with a non-zero coefficient.

Correct answer: C. \(0\)

Using the identity \((a-b)(a+b)=a^2-b^2\), we get \((x^3-4)(x^3+4)=x^6-16\). Therefore, \(P(x)=x^6-16-x^6=-16\). This is a non-zero constant polynomial, so its degree is \(0\). It is not degree \(6\), because the \(x^6\) terms cancel. Exam tip: simplify the expression and combine like terms before finding the degree.

Correct answer: C. 10

Substituting a=6 and b=10 gives the polynomial 3x^6+2x^10+1. The degree of a polynomial is the greatest exponent of x in any of its terms. The greatest exponent is 10, so its degree is 10. Although 16 is the sum of the exponents, exponents are not added to find the degree of a polynomial. Exam tip: identify the exponent of the variable in every term and select the greatest one.

Correct answer: C. \(0\)

First expand: \(x^3(x^2+x)=x^5+x^4\). Therefore, \(p(x)=x^5+x^4-x^5-x^4+15=15\). A non-zero constant polynomial such as \(15\) has degree \(0\), so \(0\) is correct. Degree \(5\) appears before simplification, but its terms cancel out. Exam tip: simplify an expression and combine like terms before finding its degree.

Correct answer: C. 5

The degree of a product of non-zero polynomials is the sum of the degrees of its factors. Here, \(x^3\) has degree 3 and \((x-2)^2\) has degree \(2\times 1=2\). Therefore, the total degree is \(3+2=5\). Option 6 is incorrect because the highest power of \(x\) in \((x-2)^2\) is only 2. Exam tip: For a polynomial written as a product of factors, add the degrees of the factors.

Correct answer: B. r=-6

A polynomial has degree 0 when it is a non-zero constant. Since 20 is already a non-zero constant term, the coefficients of both terms containing x must be zero. Thus, r+6=0, giving r=-6. If r=0, the coefficient of x^8 remains 6, so the degree would be 8. Exam tip: for degree 0, every variable term must vanish.

Correct answer: C. 9

For the product of two non-zero polynomials, the degree equals the sum of their degrees. Hence, \(\deg(p(x)q(x))=5+4=9\). Option 5 is only the degree of \(p(x)\), not of the product. Exam tip: add degrees when polynomials are multiplied; this rule does not apply directly to addition or subtraction.

Correct answer: C. 7

Find the total degree of each term: \(x^4y^2\) has degree \(4+2=6\), \(x^2y^5\) has degree \(2+5=7\), and \(xy\) has degree \(1+1=2\). The greatest of these is 7, so the total degree of the polynomial is 7. Option 6 is only the degree of the first term. Exam tip: For a term with two or more variables, add their exponents and then choose the greatest such sum.

Correct answer: B. u = −6

A polynomial has degree 0 exactly when it is a nonzero constant: every term containing a positive power of x must have zero coefficient, while the constant term must remain nonzero. Here the x^10 coefficient requires u^2−36=0, so u=6 or u=−6. The x^5 coefficient requires u+6=0, giving u=−6. The common solution is therefore u=−6. Substitution gives (36−36)x^10+0x^5+7=7, a nonzero constant, so its degree is 0. Option B is correct. At u=6, the x^5 coefficient is 12, leaving degree 5. At u=0, both variable terms remain, and u=36 also leaves a nonzero x^10 coefficient.

Correct answer: C. 2

Combining like terms gives \(3x^9-3x^9=0\) and \(4x^5-4x^5=0\). Thus, \(F(x)=6x^2-1\). The highest power of \(x\) in the simplified polynomial is 2, so its degree is 2. Option 0 applies only to a non-zero constant polynomial, but \(6x^2\) is present here. Exam tip: Always simplify a polynomial by combining like terms before finding its degree.

Correct answer: C. (4)

((x^4+2x^2)^2=x^8+4x^6+4x^4). After higher terms cancel, (4x^4+5x) remains, whose degree is (4).

Correct answer: A. (v=7)

For degree (2), both (x^{11}) and (x^6) terms must vanish. At (v=7), both coefficients become (0).

Correct answer: C. 5

The governing concept is that the degree must be determined after all like terms are combined; a leading term may cancel. Expand the product in three parts: x^2(x^5−x^4+x^3)=x^7−x^6+x^5; x(x^5−x^4+x^3)=x^6−x^5+x^4; and 1(x^5−x^4+x^3)=x^5−x^4+x^3. Adding these gives x^7+x^5+x^3 because the x^6 and x^4 terms cancel. The expression then subtracts x^7, leaving x^5+x^3. The highest nonzero exponent is 5, so option C is correct. Option A incorrectly counts the cancelled x^7 term, while options B and D overlook the actual surviving highest power.

Correct answer: C. (a=1)

For degree (3), the (x^{10}) and (x^6) terms must vanish. At (a=1), both higher-term coefficients become (0).

Correct answer: B. 7

First expand: \(x^5(x^4-3)=x^9-3x^5\). Thus, the expression becomes \(x^9-3x^5-x^9+4x^7-2=4x^7-3x^5-2\). The \(x^9\) terms cancel, and the highest remaining power of \(x\) is 7, so the degree is 7. Option 9 is the closest distractor, but its terms disappear after simplification. Exam tip: always simplify the expression completely before finding its degree.

Correct answer: D. 2

On substituting \(k=-5\), we get \(k+5=0\). Hence, both \((k+5)^2x^9\) and \((k+5)x^4\) become zero. The polynomial left is \(q(x)=-6x^2+8\). Its highest power of \(x\) is 2, so its degree is 2. Option 0 would apply only to a non-zero constant polynomial, but an \(x^2\) term remains here. Exam tip: substitute the parameter first, remove terms with zero coefficients, and then identify the highest remaining exponent.

Correct answer: B. The degree is 3 because the \(x^5\) terms cancel

On removing brackets, \(P(x)=3x^5-2x^3+x-3x^5-x^3+4=-3x^3+x+4\). The highest remaining exponent is 3, so the degree is 3. In exams, simplify and combine like terms first.

Correct answer: A. 3

The coefficient of \(x^7\) is \(3-8+5=0\), so all the \(x^7\) terms cancel. Thus, \(f(x)=2x^3-11\). The highest power of the variable in the simplified polynomial is 3, so its degree is 3. Option 7 is incorrect because the \(x^7\) terms vanish after simplification. Exam tip: Always combine like terms before finding the degree of a polynomial.

Correct answer: D. 8

For a polynomial in two variables, the total degree of a term is the sum of the exponents of x and y in that term. Here, 6x^4y^3 has degree 4+3=7, -5x^2y^6 has degree 2+6=8, 3xy has degree 1+1=2, and the constant term -2 has degree 0. The greatest total degree is 8, so the polynomial has total degree 8. Option 7 is only the degree of the first term, not of the whole polynomial. Exam tip: find the degree of every term, then choose the greatest one.

Correct answer: B. 7

To find the degree with respect to x, treat y as part of the coefficient. The powers of x in the given expression are 7, 5, 0, and 0. The greatest power is 7, so the degree is 7. Although 8 appears as a power of y, it is not the degree with respect to x. Exam tip: look only at the highest power of the variable named in the question.

Correct answer: A. 8

When the polynomial is considered with respect to \(y\), \(x\) is treated as a constant. The powers of \(y\) in the terms are \(2, 8, 3\), and \(0\), respectively. The greatest power is \(8\), occurring in \(-9x^5y^8\); hence the degree is \(8\). Option \(7\) is the highest power of \(x\), not the degree with respect to \(y\). Exam tip: for degree in one variable, compare powers of that variable only.

Correct answer: C. (n=12)

Direct answer: Option C, \\(n=12\\). The degree of a polynomial is the greatest exponent of x whose coefficient is not zero. Here the terms have exponents n, 9 and 0. Since the question gives \\(n>9\\), the exponent n is greater than 9, so the term \\(7x^n\\) has the greatest power; its coefficient 7 is nonzero. Thus the degree is n, and a value such as 12 is possible. Option A, n=8, violates n>9 and would make the x^9 term higher. Option B, n=9, also violates the strict condition n>9; it would combine like powers but does not satisfy the stated condition. Option C, n=12, satisfies the condition and gives degree 12. Option D, n=5, violates n>9 and cannot be the degree under the given statement. Exam cue: compare exponents only after checking whether the leading coefficient is zero.

Correct answer: D. \(m=6\)

The degree of a polynomial is the greatest exponent of its variable. The term \(5x^8\) ensures that the degree is 8. Since \(m<8\) is also required, \(m=6\) is possible; then the exponent in \(8x^6\) remains below 8. Although \(m=8\) would not change the degree, it does not satisfy the given inequality. Exam tip: while finding degree, always check any conditions imposed on the exponent as well.