Degree of a Polynomial

Correct answer: D. \(m=5\)

Explanation: The degree of a polynomial is the greatest exponent of the variable having a non-zero coefficient. The term \(6x^7\) makes the degree 7. Since \(m<7\), \(m=5\) is possible, and the greatest exponent still remains 7. Although \(m=7\) would also keep the degree 7, it does not satisfy the given condition \(m<7\). Exam tip: compare all exponents and then check every stated condition separately.

Correct answer: C. 9

Explanation: In \((x^3-2)^3\), the term with the highest power is \((x^3)^3=x^9\). Therefore, the highest exponent in the polynomial is 9, so its degree is 9. Option 6 is incorrect because the exponents are multiplied here: \(3\times3\), not added. Exam tip: To find the degree of a polynomial, identify the term with the greatest exponent after expansion or simplification.

Correct answer: C. \(0\)

Explanation: \((x^4+3)^2=x^8+6x^4+9\). On subtracting \((x^8+6x^4)\), the terms \(x^8\) and \(6x^4\) cancel, leaving only the non-zero constant polynomial \(9\). The degree of every non-zero constant polynomial is \(0\), so the correct answer is \(0\). Choosing \(8\) would result from ignoring the cancellation after subtraction. Exam tip: always simplify the complete expression before finding its degree.

Correct answer: B. (12)

Explanation: The degree of a polynomial is the greatest exponent of the variable whose coefficient is not zero. In this expression, the term with the greatest exponent is \\(b+2\\) times \\(x^{12}\\). The condition \\(b\\ne-2\\) guarantees that \\(b+2\\) is not zero, so this term is genuinely present. Therefore the polynomial has degree 12, and option B is correct.

The other terms, \\(5x^6\\) and \\(-4\\), have degrees 6 and 0, which are smaller than 12. A term of degree 12 cannot disappear under the given condition. If \\(b=-2\\), the leading term would vanish and the degree would become 6, but that value is excluded. Hence the highest remaining power is 12.

Correct answer: B. 6

Explanation: On substituting \(b=-2\), we get \(b+2=0\). Hence, the coefficient of \(x^{12}\) is zero, so that term disappears. The remaining polynomial is \(5x^6-4\), whose highest power of \(x\) is 6. Therefore, its degree is 6. Choosing 12 would be incorrect because the \(x^{12}\) term has zero coefficient. Exam tip: substitute the given parameter value and remove zero-coefficient terms before finding the degree.

Correct answer: B. The degree is 4 because \(5x^6-5x^6=0\)

Explanation: Combining like terms gives \(5x^6-5x^6=0\), so \(P(x)=3x^4-x+1\). The highest exponent with a non-zero coefficient is 4. Exam tip: simplify the polynomial before identifying its degree.

Correct answer: B. 5

Explanation: First expand the bracket: \(3x^5(x^2-1)=3x^7-3x^5\). Therefore, the expression becomes \(3x^7-3x^5-3x^7+8x^4=-3x^5+8x^4\). The terms \(3x^7\) and \(-3x^7\) cancel. The highest power of \(x\) in the remaining polynomial is 5, so its degree is 5. Option 7 may seem tempting before simplification, but its term cancels out. Exam tip: Always simplify a polynomial completely before finding its degree.

Correct answer: D. (0)

Explanation: Terms with zero coefficients disappear and (25) remains. It is a non-zero constant polynomial with degree (0).

Correct answer: A. (4x^8-3x^6+2)

Explanation: To identify the required polynomial, check two conditions separately: its greatest exponent must be 8, and the coefficient of \\(x^7\\) must be zero. In option A, the polynomial is \\(4x^8-3x^6+2\\). Its leading term is \\(4x^8\\), so its degree is 8. There is no \\(x^7\\) term, which means its coefficient is 0. Therefore option A satisfies both conditions.

Option B has an \\(x^7\\) term with coefficient 1, so it fails the second condition. Option C has degree 7 rather than 8, while option D has degree 9. Thus only the first polynomial has degree 8 and a zero coefficient for \\(x^7\\).

Correct answer: D. 0

Explanation: First simplify the expression:
\(x^3(x^3-2)=x^6-2x^3\). Hence,
\(p(x)=x^6-2x^3-x^6+2x^3+11=11\). This is a non-zero constant polynomial, so its degree is \(0\). Option 6 may seem likely from the initial terms, but the terms with equal powers cancel out. Exam tip: always simplify a polynomial completely before finding its degree.

Correct answer: B. \(5\)

Explanation: First, \((x^3+3x)(x^4-x)=x^7+3x^5-x^4-3x^2\). On subtracting \(x^7\), the \(x^7\) terms cancel, leaving \(3x^5-x^4-3x^2\). The highest exponent in the resulting polynomial is \(5\), so its degree is \(5\). Option \(7\) is the highest power before simplification, but it does not remain in the final polynomial. Exam tip: always simplify and combine like terms before deciding the degree.

Correct answer: A. (c=5)

Explanation: The direct answer is A, \(c=5\). A polynomial has degree 0 when all terms containing a positive power of \(x\) disappear and a nonzero constant remains. Here the coefficients of \(x^6\) and \(x^4\) must both be zero. First, \(c-5=0\) gives \(c=5\). Check the second coefficient: \(c^2-25=5^2-25=25-25=0\). The expression then becomes \(13\), whose degree is 0. Option A is correct because both variable terms vanish. Option B, \(c=-5\), makes \(c^2-25=0\), but \(c-5=-10\), so the \(x^6\) term remains and the degree is 6. Option C, \(c=0\), leaves nonzero coefficients, so the degree remains 6. Option D, \(c=25\), also leaves both relevant coefficients nonzero. Always check every positive-power term, not merely one coefficient.

Correct answer: C. 10

Explanation: The total degree of a polynomial is the greatest sum of the exponents of the variables in any one term. For x^3y^2z^5, the sum is 3+2+5=10; for 2x^4yz^3, it is 4+1+3=8; and the constant term -6 has degree 0. Hence, the total degree is 10. Option 8 is the degree of the second term, not of the whole polynomial. Exam tip: Find the sum of exponents in each term separately, then choose the greatest sum.

Correct answer: C. (6)

Explanation: A degree taken with respect to one variable counts only that variable’s exponent. Here the polynomial is being considered as a polynomial in \(z\), so expressions involving \(x\) serve as coefficients. The exponent of \(x\) does not increase the degree in \(z\). Constant terms with no \(z\) have \(z\)-degree zero.

The terms have the following powers of \(z\): \(5x^3z^6\) has power \(6\), \(-7x^2z^4\) has power \(4\), \(9x\) has power \(0\), and \(-2\) has power \(0\). The greatest power is therefore \(6\), and its coefficient \(5x^3\) is non-zero in this context. Thus option C, degree \(6\), follows.

Correct answer: C. 2

Explanation: displaystyle (x^3+x)^2=x^6+2x^4+x^2. Therefore,

displaystyle h(x)=x^6+2x^4+x^2-x^6-2x^4=x^2. The degree of a non-zero polynomial is the highest power of the variable, so the degree is 2. The terms with powers 6 and 4 cancel out. Exam tip: Expand first and combine like terms before finding the degree.

Correct answer: C. \(a=3\)

Explanation: A cubic polynomial must have highest power 3. Thus, the coefficient of \(x^5\) must be zero: \(a-3=0\), so \(a=3\). Then \(P(x)=2x^3-1\). Exam tip: ignore any term whose coefficient becomes zero.

Correct answer: A. Their leading coefficients have sum 0

Explanation: If the leading coefficients of equal-degree terms are additive inverses, their sum is 0 and the highest-degree term cancels. Hence the degree may decrease. Exam tip: check the leading coefficients first.

Correct answer: C. 2

Explanation: While removing the brackets, the sign of every term in the second polynomial changes: p(x)=x^5-3x^2+2-x^5+x^2=-2x^2+2. The highest power of x is 2, so the degree of the polynomial is 2. It is not 5 because the x^5 terms cancel each other. Exam tip: When subtracting a polynomial in brackets, change the signs of all its terms.

Correct answer: B. 5

Explanation: On expanding, \(x^5(x^2+1)-x^2(x^5-4)=x^7+x^5-x^7+4x^2=x^5+4x^2\). The \(x^7\) terms cancel out. The highest power of \(x\) in the remaining polynomial is 5, so its degree is 5. Option 7 would result from ignoring this cancellation. Exam tip: simplify a polynomial and combine like terms before finding its degree.

Correct answer: B. (8)

Explanation: As with any polynomial containing a parameter, first substitute the given value and then remove terms whose coefficients become zero. The highest displayed exponent is not automatically the degree. Here, the coefficient of \(x^{10}\) is \(e-2\), which becomes zero when \(e=2\). The next highest nonzero power is 8, so option B is correct.

Substitute \(e=2\): \((e-2)x^{10}=(2-2)x^{10}=0\), while \((e+1)x^8=(2+1)x^8=3x^8\). The polynomial becomes \(3x^8+4x^3-6\). Its greatest exponent in a nonzero term is 8, so its degree is 8. The term of degree 10 disappears, and therefore option A is not valid.

Correct answer: D. Not defined

Explanation: Every term in the given expression has coefficient 0, so it is the zero polynomial. Since the zero polynomial has no non-zero term, its degree is not defined. Option 0 is incorrect because a non-zero constant polynomial has degree 0, whereas the zero polynomial does not. Exam tip: Before finding a degree, first check whether the polynomial is the zero polynomial.

Correct answer: B. The sign of a coefficient does not affect degree; the degree of the polynomial is 7.

Explanation: The degree is the highest exponent whose coefficient is non-zero. In \(-4x^7\), the coefficient is negative but not zero, so the degree is 7. Exam tip: ignore coefficient signs and check the greatest exponent.

Correct answer: B. \(x^4y^3+2x\)

Explanation: In \(x^4y^3+2x\), the term \(x^4y^3\) has total degree \(4+3=7\). The other term, \(2x\), has degree 1, so the greatest total degree in the polynomial is 7. Hence, option B is correct. In option A, the greatest total degree is only \(2+3=5\). Exam tip: For a multivariable term, add the exponents of all variables; the largest such sum gives the total degree of the polynomial.

Correct answer: B. (13)

Explanation: Polynomial degree is determined by the largest exponent attached to a term with a nonzero coefficient. A written term with coefficient zero is not part of the effective polynomial. The condition \(t\neq0\) guarantees that the term \(tx^{13}\) cannot disappear, so its exponent must be considered when comparing the remaining terms.

The term \(0x^{15}\) contributes nothing and cannot make the degree 15. The other terms have powers 13, 6, and 0. Since \(t\neq0\), the power-13 term is present, and 13 is the greatest nonzero power. Therefore the degree is 13, so option B is correct. Choosing 15 would incorrectly treat a zero term as an actual term.

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

Explanation: If the leading terms of non-zero polynomials are \(ax^m\) and \(bx^n\), their product has leading term \(abx^{m+n}\). Hence its degree is \(m+n\), not the larger degree. Exam tip: note the non-zero condition.