Degree of a Polynomial

Correct answer: D. 8

Explanation: The degree of a polynomial is the greatest exponent of the variable in any term with a non-zero coefficient. Here, the exponents of x are 2, 4, 6, and 8. The greatest exponent is 8, so the degree of the polynomial is 8. Option 6 is a close distractor, but the presence of the x^8 term means that the degree cannot be 6. Exam tip: identify the term with the highest power and a non-zero coefficient before deciding the degree.

Correct answer: A. 7

Explanation: The degree of a polynomial is the greatest exponent of the variable in any term. Here, the term \(-y^7\) has exponent 7, so the degree is 7. Choosing 4 from the first term is incorrect. Exam tip: check every term for the highest power.

Correct answer: B. 10

Explanation: The degree of a polynomial in one variable is the highest exponent of the variable having a non-zero coefficient. Here, the exponents of x are 10, 7, 2, and 0; the greatest is 10. Therefore, the degree of the polynomial is 10. The number 3 is only the coefficient of x^{10}, not the degree. Exam tip: treat a constant term as having exponent 0 and look for the greatest exponent.

Correct answer: C. 7

Explanation: The degree of a polynomial is the greatest exponent of the variable among terms with non-zero coefficients. Here the exponents are n, 3, and 0. Since n > 3, the greatest exponent is n. As the given degree is 7, n = 7. Choosing 3 would also violate the condition n > 3. Exam tip: a non-zero constant term has degree 0.

Correct answer: A. (a) can be any real number

Explanation: The degree of a polynomial is determined by the highest power whose coefficient is not zero. In the expression x^4+ax^2+7, the coefficient of x^4 is fixed as 1. Since 1 is never zero, the x^4 term is always present, regardless of the value chosen for a. The middle term may disappear, but that cannot lower the degree below 4.

If a=0, the expression becomes x^4+7 and still has degree 4. If a is non-zero, the x^2 term is present, but x^4 remains the highest power. Thus every real value of a gives degree 4. Option A is correct. The value a=4 is not required, and a≠0 is unnecessary because zero also works.

Correct answer: A. \(5x^2-9\)

Explanation: In \(5x^2-9\), the highest power of \(x\) is \(2\), so its degree is \(2\). There is no \(x\)-term, which can be written as \(0x\); therefore, the coefficient of \(x\) is \(0\). In \(5x^2+x-9\), the coefficient of \(x\) is \(1\), so it is not correct. Exam tip: the coefficient of an absent term is always \(0\).

Correct answer: C. 5

Explanation: Expanding the expression gives \(x^3(2x^2-5)+4x^2=2x^5-5x^3+4x^2\). The highest power of \(x\) is 5, and its coefficient, 2, is non-zero. Therefore, the degree of the polynomial is 5. The outside factor \(x^3\) multiplies \(2x^2\), so their exponents add to give \(x^5\); hence 3 is not the degree. Exam tip: expand brackets and simplify terms before identifying the highest exponent.

Correct answer: B. 3

Explanation: The degree of a polynomial is the greatest exponent of the variable whose coefficient is nonzero after all given values have been substituted and the expression has been simplified. A term with a zero coefficient disappears and cannot determine the degree. Therefore, it is necessary to substitute s = 1 before inspecting the powers of x.

The expression becomes (1 − 1)x⁴ + (1 + 3)x³ + 2, which simplifies to 0x⁴ + 4x³ + 2, or 4x³ + 2. The fourth-degree term vanishes, but the coefficient of x³ is 4, which is nonzero. Hence the greatest remaining exponent is 3, so option B is correct. The answer is not 4 because the x⁴ term has disappeared, and the constant term does not control the degree.

Correct answer: C. 5

Explanation: The total degree of a polynomial is the greatest sum of the exponents of variables in any term. Here, the degree of 2x^2y is 2+1=3, the degree of 3xy^4 is 1+4=5, and the constant 5 has degree 0. Therefore, the total degree is 5. Option 3 is only the degree of the first term. Exam tip: Add the exponents in each term and select the greatest sum.

Correct answer: C. \(6x^4+7\)

Explanation: The degree of a polynomial is the greatest exponent of its variable with a non-zero coefficient. In \(6x^4+7\), the highest power of \(x\) is 4, so its degree is 4, not 5. Each of the other options contains an \(x^5\) term and therefore has degree 5. Exam tip: For degree, look for the highest power of the variable, not the constant term.

Correct answer: C. 6

Explanation: The degree of a polynomial is the greatest power of the variable with a non-zero coefficient. Here, \(v\neq2\), so \(v-2\neq0\). Thus, the coefficient of \(x^6\) is non-zero, and the highest power of \(x\) is \(6\). Therefore, the degree is \(6\). Option \(4\) would be correct only if \(v=2\), making the \(x^6\) term vanish. Exam tip: In a polynomial involving a parameter, check the coefficient of the highest-power term first.

Correct answer: A. 9

Explanation: The expression 9 contains no variable. It is a non-zero constant polynomial, so its degree is 0. The degrees of 4x and x+9 are 1, while that of 2x^2+1 is 2; having a constant term does not make 2x^2+1 a constant polynomial. Exam tip: the highest power of the variable gives the degree of a polynomial.

Correct answer: B. (5)

Explanation: The direct answer is option B, 5. The degree of a polynomial is the greatest exponent of the variable whose coefficient is not zero. Substitute \(w=3\): \((w-3)x^8=(3-3)x^8=0x^8\), so the eighth-degree term disappears. Also, \((w+1)x^5=(3+1)x^5=4x^5\). The expression becomes \(4x^5-2x+6\), whose highest nonzero power is 5. Option A, 8, is wrong because the coefficient of \(x^8\) becomes zero. Option B, 5, is correct. Option C, 1, ignores the nonzero \(x^5\) term and notices only the linear term. Option D, 0, would apply only to a nonzero constant polynomial, not this expression. Memory cue: a zero coefficient removes a term before finding degree.

Correct answer: A. \(7x^3-2x+5\)

Explanation: In \(7x^3-2x+5\), the highest power of x is 3, and all powers of x are non-negative integers. Hence, it is a polynomial in x of degree 3. \(x^3+\frac{1}{x}\) contains \(x^{-1}\), while \(\sqrt{x}+x^3\) contains \(x^{1/2}\), so neither is a polynomial. \(x^4-1\) is a polynomial, but its degree is 4. Exam tip: in a polynomial, powers of the variable must be non-negative integers such as 0, 1, 2, and 3.

Correct answer: D. m = 2

Explanation: The degree of a polynomial is the greatest exponent attached to a term with a non-zero coefficient. The term 7x^4 is already present, so the polynomial has degree at least 4. The condition m < 4 requires the exponent in 2x^m to be strictly smaller than 4; consequently, that term cannot create a degree greater than 4. Among the choices, only m = 2 satisfies the strict inequality, and the highest exponent remains 4. Therefore, option D is correct. Values m = 6 and m = 5 violate m < 4 and would raise the degree. Although m = 4 would preserve degree 4, it fails the stated strict condition.

Correct answer: A. डिग्री सबसे बड़े गुणांक से नहीं, चर के सबसे बड़े घातांक से निर्धारित होती है; अतः डिग्री 5 है।

Explanation: The degree of a polynomial is the greatest exponent of the variable among its non-zero terms, not the largest coefficient. Here the exponents are 0, 2, and 5, so the degree is 5. Exam tip: list exponents first, then choose the maximum.

Correct answer: C. 5

Explanation: The coefficient of \(0x^8\) is 0, so it does not contribute to the polynomial. Among the remaining non-zero terms, \(7x^5\), \(-4x^2\), and \(1\), the highest exponent is 5. Therefore, the degree of the polynomial is \(5\). Option 8 is incorrect because a term with zero coefficient does not determine the degree. Exam tip: Remove all zero-coefficient terms before finding the degree.

Correct answer: B. 3

Explanation: The terms 6x^4 and -6x^4 are like terms, and their sum is 0. Thus, the expression simplifies to 3x^3-2. Its highest-power term is 3x^3, so the degree of the polynomial is 3. Option 4 is incorrect because the x^4 terms cancel out. Exam tip: Always simplify like terms before finding the degree of a polynomial.

Correct answer: C. 3

Explanation: Substituting a=-2 gives a+2=0, so the x^7 term disappears. The polynomial becomes p(x)=4x^3-5. Its highest power of x is 3, so its degree is 3. Choosing 7 would be incorrect because its coefficient becomes zero. Exam tip: Substitute the given value first and remove terms with zero coefficients before finding the degree.

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

Explanation: The degree of a polynomial is the highest power of a variable whose coefficient is non-zero. For the degree to be 2, the \(x^5\) term must vanish, so \(m-6=0\). Hence, \(m=6\), and the polynomial becomes \(2x^2+7\), which has degree 2. For a value such as \(m=-6\), the coefficient of \(x^5\) is not zero, so the degree remains 5. Exam tip: In a polynomial involving a parameter, first check the coefficient of the highest-power term.

Correct answer: A. (-9x^7)

Explanation: A polynomial’s degree is found by identifying the largest exponent attached to the variable in a nonzero term. The coefficient may be positive or negative, but that sign does not affect the exponent. Constant terms have exponent zero, so they are considered only when no variable term has a larger power. Here, the term with the greatest power is \(-9x^7\), making option A correct.

The powers in the polynomial are 7, 5, 1, and 0. Since \(-9\) is nonzero, the term \(-9x^7\) is present and its exponent is 7. The other variable terms have smaller exponents, and 8 is a constant. Thus the degree is 7 and is decided by \(-9x^7\). A negative coefficient does not make the term invalid and does not lower its degree.

Correct answer: D. 10

Explanation: The degree of a polynomial is the greatest exponent of the variable in any term with a non-zero coefficient. Here, the exponents are 2, 10, 4, and 0. The greatest exponent is 10, so the degree of the polynomial is 10. Although 4 is an exponent of one term, it is not the degree of the whole polynomial. Exam tip: The order of terms does not matter; identify the highest power of the variable.

Correct answer: C. 1

Explanation: Terms with coefficient 0 do not contribute to a polynomial. Thus, q(x)=0x^6+0x^4-13x+9 simplifies to -13x+9. The highest exponent of x in this polynomial is 1, so its degree is 1. Degrees 6 and 4 are not considered because the coefficients of x^6 and x^4 are zero. Exam tip: First remove all terms with zero coefficients before finding the degree.

Correct answer: C. 5

Explanation: Since \(x^0=1\), \(2x^0=2\). The polynomial can be written as \(9x^5-4x^2+2\). The greatest exponent of \(x\) with a non-zero coefficient is \(5\), so the degree of the polynomial is \(5\). The value \(2\) is only the degree of one term, not of the whole polynomial. Exam tip: simplify terms such as \(x^0\) first, then identify the highest exponent.

Correct answer: B. 0

Explanation: (-24) is a non-zero constant polynomial because it contains no variable term. The degree of every non-zero constant polynomial is 0. A polynomial of degree 1 must have a variable with highest power 1, which is absent here. Exam tip: only the zero polynomial has an undefined degree; a non-zero constant has degree 0.