Degree of a Polynomial

On subtraction, the like terms cancel: \(A(x)-B(x)=(x^8+4x^3+7)-(x^8+4x^3)=7\). Since \(7\) is a non-zero constant polynomial, its degree is \(0\). Degree \(8\) is the degree of the original polynomials, not of their difference. Exam tip: simplify a sum or difference of polynomials before deciding its degree.

\(A(x)+B(x)=(x^7+5)+(-x^7+6x^4)=6x^4+5\). The \(x^7\) and \(-x^7\) terms cancel each other. The highest power in the remaining polynomial is \(4\), so its degree is \(4\). Option \(7\) is not correct because no \(x^7\) term remains after addition. Exam tip: first simplify the sum, then identify the highest power with a non-zero coefficient.

Using the identity \((a-b)(a+b)=a^2-b^2\), we get \((x^4-5)(x^4+5)=x^8-25\). Therefore, \(P(x)=x^8-25-x^8=-25\). Since this is a non-zero constant polynomial, its degree is \(0\). Option \(8\) may seem tempting before simplification, but the \(x^8\) terms cancel. Exam tip: always simplify a polynomial completely before identifying its degree.

Substituting the given values, the polynomial becomes 5x^7+3x^11+1. The degrees of its terms are 7, 11, and 0. The greatest exponent of x with a non-zero coefficient is 11, so the degree of the polynomial is 11. Option 18 is the sum of the exponents, but the degree of a polynomial is not found by adding exponents of different terms. Exam tip: identify the exponent of the variable in every term and select the greatest one.

On multiplying, \(x^4(x^3+x)=x^7+x^5\). Hence, \(p(x)=x^7+x^5-x^7-x^5+23=23\). A non-zero constant polynomial has degree \(0\), so the correct answer is \(0\). Degree \(7\) is only the highest exponent visible before simplification; those terms cancel. Exam tip: always simplify a polynomial completely before finding its degree.

\((x-3)^2\) has degree 2, and \(x^4\) has degree 4. The degrees of two non-zero polynomials add when they are multiplied, so the degree of \(x^4(x-3)^2\) is \(4+2=6\). Option 8 is incorrect because the exponents are not multiplied. Exam tip: For a polynomial written as a product, add the degrees of its factors.

A polynomial of degree 0 is a non-zero constant, so the coefficients of both the \(x^9\) and \(x^2\) terms must be zero. Both coefficients are \(r-8\). Thus, \(r-8=0\), giving \(r=8\). For this value, the polynomial becomes \(14\), whose degree is 0. If \(r=9\), the coefficient of \(x^9\) remains non-zero, so the degree would be 9. Exam tip: In degree-reduction questions, first make the coefficient of the highest-power term zero.

For non-zero polynomials, the degree of their product equals the sum of their degrees. Thus, deg(p(x)q(x)) = 6 + 5 = 11. Option 6 is only the degree of p(x), not of the product. Exam tip: add degrees when polynomials are multiplied; for addition or subtraction, generally consider the larger degree.

The total degree of a polynomial is the greatest sum of the exponents of the variables in any one term. Here, the total degrees of x^5y^2, x^3y^6, and xy are 5+2=7, 3+6=9, and 1+1=2 respectively. The greatest value is 9, so option C is correct. Option A is the degree of the first term, not of the whole polynomial. Exam tip: find the sum of exponents in each term and select the greatest one.

A polynomial has degree 0 when every positive-power term disappears and a non-zero constant remains. Therefore both coefficients must be zero: u^2−64=0 for the x^11 term and u+8=0 for the x^5 term. The second equation immediately gives u=−8. Checking it in the first equation, (−8)^2−64 = 64−64 = 0, so both variable terms vanish. The expression then reduces to the non-zero constant 3, whose degree is 0. Hence option B is correct. Although u=8 also makes u^2−64 zero, it gives u+8=16, leaving 16x^5; its degree would therefore be 5, not 0. The other options fail as well.

In the given expression, 4x^{10}-4x^{10}=0 and 5x^6-5x^6=0. Thus, F(x)=7x^3-2 remains. The greatest exponent of x is 3, so the degree of the polynomial is 3. Degree 0 would apply only to a non-zero constant polynomial. Exam tip: simplify like terms first, then identify the highest remaining exponent.

((x^5+3x^2)^2=x^{10}+6x^7+9x^4). After higher terms cancel, (13x^4) remains, whose degree is (4).

For degree (4), both (x^{12}) and (x^8) terms must vanish. At (w=9), both coefficients become (0).

For a multivariable polynomial, first find the total degree of each term by adding the exponents of all variables in that term. The term x^5y^2z^4 has total degree 5+2+4=11. In 3x^3yz^7, the coefficient 3 does not affect degree and the unmarked y has exponent 1, so its total degree is 3+1+7=11. The constant term −10 has degree 0. The degree of the entire polynomial is the greatest total degree among its terms, which is 11. Therefore option B is correct. Option A may result from omitting one exponent, option C comes from an incorrect addition, and option D is not the total degree of any term in the expression.

A polynomial has degree 0 when it reduces to a nonzero constant. Therefore both positive-power terms must vanish: u² − 81 = 0 for the x^13 term and u − 9 = 0 for the x^6 term. The second condition immediately gives u = 9. Checking it in the first condition, 9² − 81 = 81 − 81 = 0, so both variable terms disappear and the expression becomes the nonzero constant 5. Its degree is therefore 0, making option A correct. Although u = −9 satisfies u² − 81 = 0, it gives u − 9 = −18, so a nonzero x^6 term remains and the degree is 6. Values 0 and 81 satisfy neither required pair of conditions.

The governing concept is that polynomial multiplication uses the distributive law, and the degree must be found after combining like terms and cancelling any equal terms. Expand the product: x^7·x^8=x^15, x^7·(−x^4)=−x^11, 4x^3·x^8=4x^11, and 4x^3·(−x^4)=−4x^7. Thus the product is x^15+3x^11−4x^7. Subtracting x^15 cancels the two x^15 terms, leaving 3x^11−4x^7. The coefficient 3 is nonzero, so the highest remaining exponent is 11 and the degree is 11. Choosing 15 overlooks the cancellation; 8 and 7 do not represent the largest remaining exponent. Therefore option B is correct.

For degree (6), the (x^{10}) term must vanish and the (x^6) term must remain. This condition holds at (a=4).

When an expression contains brackets, it must first be expanded and like terms must then be combined. This is essential because the terms with the greatest visible powers may cancel. The degree is determined only after all such cancellation has been completed.

Distribute \(2x^5\): \(2x^5(x^3-1)=2x^8-2x^5\). The full expression becomes \(2x^8-2x^5-2x^8+7x^4\). The two degree-8 terms cancel because their coefficients are 2 and \(-2\). The remaining expression is \(-2x^5+7x^4\), whose highest exponent is 5. Therefore choice B, (5), is correct.

On substituting k=-1, we get (k+1)^2=0 and k^2-1=0. Thus, the x^9 and x^5 terms vanish, leaving q(x)=-4x+6. The highest power of x in the remaining polynomial is 1, so its degree is 1. Degree 0 would apply only to a non-zero constant polynomial. Exam tip: substitute the parameter first, remove terms with zero coefficients, and then identify the highest remaining exponent.

The degree is the greatest exponent of \(x\) with a non-zero coefficient. Thus, an \(x^4\) term must remain and no higher power may occur. Exam tip: count the highest valid exponent, not the number of terms.

For non-zero polynomials, the degree of a product equals the sum of their degrees: \(5+5=10\). In a sum or difference, leading terms may cancel. Exam tip: add degrees only for multiplication.

The total degree of a polynomial is the greatest sum of the exponents of the variables in any one term. Here, the total degrees of the terms are 5+3=8, 2+8=10, 1+1=2, and 0 for the constant term. The greatest value is 10, so the correct answer is 10. Note that 8 is only the total degree of the first term, 4x^5y^3. Exam tip: Add the exponents in each term and select the largest sum.

To find the degree with respect to x, treat y as part of the coefficient. The powers of x in the given expression are 6, 4, 0, and 0. The greatest power is 6, so the degree of the polynomial with respect to x is 6. The number 9 is the power of y, not the degree with respect to x. Exam tip: Look only at the highest exponent of the variable named in the question.

To find the degree with respect to y, consider only the powers of y; x is treated as part of the coefficient. The powers of y in the expression are 9, 2, 1, and 0. The greatest power is 9, so the degree is 9. Although 6 is the greatest power of x, it is not the degree with respect to y. Exam tip: always identify the variable named in the question before selecting the highest exponent.

The degree of a polynomial is the greatest exponent of its variable. The term \(4x^7\) makes the degree 7. Since \(m<7\), \(m=5\) is possible, and the greatest exponent still remains 7. \(m=7\) does not satisfy the strict inequality, while \(m=8\) or \(m=10\) would make the polynomial’s degree 8 or 10. Exam tip: compare the exponents of all variable terms to find the degree.