Degree of a Polynomial

The degree of \(x^5\) is 5, and the degree of \((x-2)^2\) is 2. The degree of the product of two non-zero polynomials is the sum of their degrees, so the total degree is \(5+2=7\). Option \(10\) may result from incorrectly multiplying the degrees. Exam tip: add degrees in a product; multiply the degree only when a polynomial itself is raised to a power.

A polynomial has degree 0 when it reduces to a non-zero constant. Therefore, the coefficients of both the \(x^8\) and \(x^3\) terms must be zero. Both coefficients are \(r-7\), so \(r-7=0\), giving \(r=7\). If \(r=-7\), the higher-power terms do not vanish. Exam tip: In a parameter-based polynomial, first set the coefficient of the highest power to zero when checking whether the degree can decrease.

For the product of two non-zero polynomials, the degree equals the sum of their degrees. Hence, \(\deg(p(x)q(x))=4+6=10\). Option \(4\) is only the degree of \(p(x)\), while \(24\) results from multiplying the degrees, which is not the correct rule. Exam tip: add degrees when polynomials are multiplied; do not multiply them.

The total degree of a polynomial is the greatest total degree among its terms. Here, the total degrees of x^4y^3, x^2y^6, and xy are 4+3=7, 2+6=8, and 1+1=2 respectively. The greatest value is 8, so the correct answer is 8. Option 7 is only the total degree of the first term. Exam tip: For each term with two or more variables, add the exponents and choose the largest sum.

A polynomial has degree 0 when every term containing a positive power of the variable disappears, while a nonzero constant remains. Thus both variable-term coefficients must be zero. For x^10, require u^2 − 49 = 0, which gives u = 7 or u = −7. For x^4, require u − 7 = 0, which gives u = 7. The common value is therefore u = 7. Substitution confirms the result: (49 − 49)x^10 + (7 − 7)x^4 + 5 = 5, and the nonzero constant 5 has degree 0. Hence option A is correct. At u = −7, the x^4 coefficient is −14, so a degree-four term remains. The other options also fail to remove both variable terms.

On combining like terms, \(3x^{11}-3x^{11}=0\) and \(4x^7-4x^7=0\). Thus, \(F(x)=5x^2-1\). The highest exponent of the variable in the simplified polynomial is 2, so its degree is 2. Option 11 would result from noticing the original highest power without first cancelling the like terms. Exam tip: Always simplify a polynomial before finding its degree.

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

For degree (2), both (x^{12}) and (x^6) terms must vanish. At (v=8), both coefficients become (0).

The governing idea is that the degree must be found after expansion and cancellation, not merely by looking at the largest exponent in an intermediate product. Multiply each part of the first factor: x^3 times the second factor gives x^9 − x^8 + x^7; x^2 times it gives x^8 − x^7 + x^6; and 1 times it gives x^6 − x^5 + x^4. Adding these expressions cancels −x^8 with x^8 and −x^7 with x^7, leaving x^9 + 2x^6 − x^5 + x^4. The external −x^9 then cancels x^9, so the final polynomial is 2x^6 − x^5 + x^4. Its highest nonzero exponent is 6, making option C correct. Option A ignores the final cancellation, and B or D do not represent the highest surviving power.

The governing concept is that a polynomial should be simplified before its degree is identified, because terms of equal powers may cancel. Distribute x^3 through the bracket: x^3·x^5 = x^8, x^3·(−2x^3) = −2x^6, and x^3·1 = x^3. Thus T(x) = x^8 − 2x^6 + x^3 − x^8 + 2x^6 + 5x. The x^8 terms cancel exactly, and the −2x^6 and +2x^6 terms also cancel. The simplified expression is x^3 + 5x. Its highest exponent with a nonzero coefficient is 3, so option C is correct. Options A and B count terms that disappear, whereas option D gives the lower exponent rather than the polynomial’s degree.

For degree (4), both (x^{11}) and (x^7) terms must vanish. At (w=-5), both coefficients become (0).

The key principle is to determine the highest nonzero power only after all like terms have been combined. Expand the product in two parts. From x^4 we obtain x^9 − x^6 + x^4, and from x we obtain x^6 − x^3 + x. Adding them gives x^9 + x^4 − x^3 + x because −x^6 and +x^6 cancel. The separate term −x^9 then cancels the remaining x^9 term. The resulting polynomial is x^4 − x^3 + x, whose highest nonzero exponent is 4. Therefore option C is correct. Option A counts the x^9 term before cancellation, and option B counts the x^6 terms even though they cancel each other. Option D is only the smallest positive exponent remaining.

The direct answer is option C, 3. Begin by expanding the square: (x^6−2)^2=x^12−4x^6+4. Substitute this into M(x): M(x)=x^12−4x^6+4−(x^12−4x^6)+9x^3. The x^12 terms cancel, and the −4x^6 terms also cancel. Thus M(x)=9x^3+4. Its highest non-zero power is x^3, so its degree is 3. Option A, 12, ignores the cancellation of the x^12 terms. Option B, 6, ignores that the x^6 terms also cancel. Option C is correct because 9x^3 remains and 9 is non-zero. Option D, 0, would apply only to a non-zero constant polynomial, but 9x^3 is present. The important lesson is that degree must be found after simplifying completely; one must not choose the largest exponent visible in the unsimplified expression. Memory cue: expand, combine like terms, cancel, and only then read the highest power.