Degree of a Polynomial

Correct answer: C. \(-7x^4+3x^2-x+9\)

Explanation: In option C, the highest power of x is 4 and every exponent is a non-negative integer, so its degree is 4. A has \(x^{-1}\), D has \(x^{1/2}\), and B has degree 5. Exam tip: check the highest valid exponent first.

Correct answer: A. Its degree is 0.

Explanation: A non-zero constant polynomial, such as 7 or -3, can be written as 7x^0 or -3x^0. Its highest power of x is 0, so its degree is 0. The zero polynomial is a separate case; its degree is undefined. Exam tip: first check whether the constant is zero.

Correct answer: D. 7

Explanation: On substituting b=4, we get b-4=0. Hence, the x^{13} term becomes 0x^{13} and disappears, leaving the polynomial 6x^7-2. The highest power of x in the remaining polynomial is 7, so its degree is 7. Choosing 13 would be incorrect because its coefficient has become zero. Exam tip: check coefficients first; a term with zero coefficient is not considered while finding degree.

Correct answer: A. \(\deg(pq)=\deg p+\deg q\)

Explanation: The product of the leading terms of two non-zero polynomials is non-zero, so the degree of their product is the sum of their degrees. The maximum-degree rule can apply to addition. Exam tip: add degrees for a product.

Correct answer: B. 6

Explanation: Expanding gives 4x^6(x^2-1)=4x^8-4x^6. Therefore, the expression becomes 4x^8-4x^6-4x^8+3x^5=-4x^6+3x^5. The x^8 terms cancel, and the highest remaining power of x is 6, so the degree is 6. Option 8 is incorrect because no x^8 term remains after simplification. Exam tip: Expand brackets and combine like terms before finding a polynomial’s degree.

Correct answer: D. (0)

Explanation: Terms with zero coefficients disappear and (-31) remains. It is a non-zero constant polynomial with degree (0).

Correct answer: A. (5x^9-2x^7+1)

Explanation: The required polynomial must have degree 9 and must not contain an \\(x^8\\) term. Option A is \\(5x^9-2x^7+1\\). Its highest power is 9, with nonzero coefficient 5, so its degree is 9. The powers shown are 9, 7, and 0; there is no \\(x^8\\) term. Therefore the coefficient of \\(x^8\\) is 0, and option A is correct.

Option B includes \\(x^8\\) with coefficient 1, so it fails the second condition. Option C has degree 8, not 9. Option D has an \\(x^{10}\\) term and therefore degree 10. Checking the highest power and the missing power identifies A uniquely.

Correct answer: C. Not defined

Explanation: Here, \(H(x)=0\) is the zero polynomial, meaning that all its coefficients are zero. It has no non-zero term with a highest power, so its degree is not defined. Option 0 is the value of the zero polynomial, not its degree. Exam tip: While finding degree, look for the greatest power among terms with non-zero coefficients.

Correct answer: D. 0

Explanation: First simplify the expression: \(x^4(x^2-4)=x^6-4x^4\). Thus, \(p(x)=x^6-4x^4-x^6+4x^4+17=17\). This is a non-zero constant polynomial, so its degree is \(0\). Choosing \(6\) or \(4\) would ignore the cancellation of like terms. Exam tip: always simplify a polynomial completely before finding its degree.

Correct answer: A. 6

Explanation: On expanding and simplifying, \((x^4+2x)(x^5-x)-x^9=x^9-x^5+2x^6-2x^2-x^9=2x^6-x^5-2x^2\). The \(x^9\) terms cancel each other. The highest power in the remaining polynomial is \(6\), so its degree is 6. Option 9 is incorrect because the \(x^9\) term does not remain after simplification. Exam tip: Always expand and combine like terms before finding the degree.

Correct answer: B. (c=-4)

Explanation: The direct answer is option B, \(c=-4\). A polynomial has degree 0 when every term containing a positive power of \(x\) disappears, while the constant term remains non-zero. Here the coefficients of \(x^7\) and \(x^5\) are \(c+4\) and \(c^2-16\). Put \(c=-4\): \(c+4=-4+4=0\), and \(c^2-16=(-4)^2-16=16-16=0\). The expression becomes \(21\), whose degree is 0. Option A, \(c=4\), makes \(c+4=8\), so an \(x^7\) term remains. Option B works because both variable terms vanish. Option C, \(c=0\), gives non-zero coefficients and degree 7. Option D, \(c=16\), also leaves the \(x^7\) term. Memory cue: for degree 0, cancel every positive-power term, then check that the constant is not zero.

Correct answer: A. (d=3)

Explanation: Direct answer: Option A, \\(d=3\\). For the polynomial to have degree 4, the x^6 term must disappear, while the x^4 term must remain nonzero. The coefficient of x^6 is d-3, so set d-3=0, giving d=3. Then the x^4 coefficient is d+1=4, which is nonzero, and the polynomial becomes \\(4x^4+12\\), whose degree is 4. Option A works for both requirements. Option B, d=-1, removes the x^4 term but leaves coefficient -4 on x^6, so the degree is 6. Option C, d=0, leaves coefficients -3 and 1, so the degree is 6. Option D, d=6, leaves coefficient 3 on x^6, again giving degree 6. The key is that cancelling the highest term is not enough; the next term must survive. Memory cue: desired degree means higher powers vanish and the desired power remains.

Correct answer: C. (9)

Explanation: For a polynomial in several variables, the total degree of a term is found by adding the exponents of all variables in that term. The degree of the whole polynomial is the greatest total degree among its nonzero terms. A constant such as \(-8\) has degree 0 and cannot exceed the degrees of the variable terms here.

For \(x^4y^2z^3\), the total degree is \(4+2+3=9\). For \(5x^2yz^6\), the exponent of y is 1, so its total degree is \(2+1+6=9\). The constant has degree 0. The greatest total degree is therefore 9, making option C correct. One must add exponents within a term, not choose only the largest individual exponent.

Correct answer: C. (7)

Explanation: When a polynomial is considered with respect to \(z\), the degree is found by comparing only the exponents of \(z\). Any power of \(x\) is part of the coefficient from this viewpoint. Terms without \(z\), such as \(3x^4\) and \(-9\), have degree zero in \(z\), regardless of the power of \(x\) appearing in them.

In \(6x^2z^7-4xz^5+3x^4-9\), the powers of \(z\) are \(7\), \(5\), \(0\), and \(0\), respectively. The first term has the greatest \(z\)-power, namely \(7\), and its coefficient \(6x^2\) is non-zero. Therefore the degree with respect to \(z\) is \(7\), which is option C. It would be incorrect to choose \(9\) by combining exponents from different variables.

Correct answer: C. 2

Explanation: To determine the degree, the square must be expanded and like powers must be combined before the highest exponent is selected. Using (a + b)² = a² + 2ab + b² with a = x⁴ and b = x gives (x⁴ + x)² = x⁸ + 2x⁵ + x². Substituting this into h(x) gives h(x) = x⁸ + 2x⁵ + x² − x⁸ − 2x⁵. The x⁸ terms cancel each other, and the 2x⁵ terms also cancel each other. Thus h(x) = x². Its only nonzero term has exponent 2, so the degree is 2 and option C is correct. Options A and B are distractors because those powers appear temporarily but disappear after cancellation.

Correct answer: A. The degree of \(p(x)+q(x)\) will be 4

Explanation: Since \(q(x)\) cannot contain an \(x^4\) term, it cannot cancel the leading \(x^4\) term of \(p(x)\). Hence their sum has degree 4. Exam tip: for polynomials of unequal degrees, the sum has the higher degree.

Correct answer: B. The degree is 4 because the highest exponent of \(x\) with a non-zero coefficient is 4.

Explanation: The degree depends on the greatest exponent of the variable having a non-zero coefficient, not on the number of terms. Here the highest power is \(x^4\), so the degree is 4. Exam tip: ignore coefficients while finding degree.

Correct answer: B. 3

Explanation: On removing the brackets, every term of the second polynomial changes sign: \(p(x)=x^6-4x^3+3-x^6+x^3=-3x^3+3\). The \(x^6\) terms cancel, and the highest power in the remaining polynomial is 3. Therefore, the degree is 3. Choosing 6 is incorrect because no \(x^6\) term remains after simplification. Exam tip: always simplify a polynomial completely before finding its degree.

Correct answer: A. The coefficient of \(x^4\) is non-zero, and there is no term with power greater than 4.

Explanation: The degree of a non-zero polynomial is the greatest exponent whose coefficient is non-zero. Thus, a non-zero \(x^4\) term with no higher-power term gives degree 4. If an \(x^5\) term exists, the degree is at least 5. Exam tip: ignore terms with zero coefficients.

Correct answer: D. Not defined

Explanation: The given expression is p(x)=0 because the coefficient of every power of x is 0. Hence, it is the zero polynomial. Since the zero polynomial has no non-zero term, its degree is not defined. Choosing 8 is incorrect because the coefficient of x^8 is also 0, so it is not an effective term. Exam tip: While finding degree, consider only terms with non-zero coefficients.

Correct answer: C. \(\deg[p(x)^2]=14\)

Explanation: For non-zero polynomials, degrees add on multiplication. Hence \(\deg[p(x)^2]=7+7=14\). But \(p(x)-p(x)\) is the zero polynomial, whose degree is not defined. Exam tip: add degrees when polynomials are multiplied.

Correct answer: B. \(x^5y^3+2x\)

Explanation: The total degree of a polynomial is the greatest sum of exponents of variables in any term. In \(x^5y^3+2x\), the term \(x^5y^3\) has total degree \(5+3=8\), so the polynomial has total degree 8. Options A and D have total degree 5, while option C has degree 6. Exam tip: For a term with more than one variable, add the exponents to find its total degree.

Correct answer: B. (14)

Explanation: The degree is the largest exponent whose coefficient is nonzero. The term \(0x^{16}\) is equal to zero, so it is omitted when determining the degree. The condition \(t\neq0\) ensures that \(tx^{14}\) is a genuine term rather than another zero term.

After ignoring \(0x^{16}\), the polynomial has relevant powers 14, 7, and 0. Since the coefficient of \(x^{14}\) is nonzero, the largest surviving exponent is 14. Therefore the degree is 14, so option B is correct. Option A would be correct only if the coefficient of \(x^{16}\) were nonzero; here it is explicitly zero.

Correct answer: C. 0

Explanation: On opening the brackets, the like terms cancel:

((x^6+x^3+1)-(x^6+x^3)) = 1.

Since 1 is a non-zero constant polynomial, its degree is 0. Although 6 and 3 are exponents in the original expression, those terms do not remain after simplification. Exam tip: Always simplify a polynomial completely before finding its degree.

Correct answer: B. (s=-3)

Explanation: For degree (0), both (x^{10}) and (x^6) terms must vanish. At (s=-3), both coefficients become (0).