Degree of a Polynomial

Correct answer: C. \(4\)

Explanation: Since \(x^0=1\), the polynomial becomes \(1+6x^4-3x^2\). The highest exponent of \(x\) with a non-zero coefficient is \(4\), so its degree is \(4\). \(2\) is only the exponent in the term \(-3x^2\), not the greatest exponent. Exam tip: simplify terms such as \(x^0\) into constants before finding the degree.

Correct answer: C. \(0\)

Explanation: The polynomial \(15\) is a non-zero constant polynomial because it contains no variable term. The degree of every non-zero constant polynomial is \(0\). Degree \(1\) is for linear polynomials, not constant polynomials. Exam tip: Any non-zero fixed number, such as \(7\), \(-3\), or \(15\), has degree \(0\) when treated as a polynomial.

Correct answer: C. Not defined

Explanation: The degree of a non-zero polynomial is the greatest exponent with a non-zero coefficient. The zero polynomial has no non-zero coefficient and therefore has no greatest exponent under the usual school-level definition. Its degree is consequently considered not defined. It should not be confused with a non-zero constant polynomial such as 5, whose degree is 0; “infinite” is not the standard answer in this context.

Correct answer: C. 5

Explanation: Combining like terms gives 3x^5+2x^5=5x^5, so the polynomial becomes 5x^5-4x^2+8. The highest power of x is 5; therefore, its degree is 5. The number 8 is a constant term, so it does not make the degree 8. Exam tip: simplify a polynomial first, then identify the greatest exponent of the variable with a non-zero coefficient.

Correct answer: C. 7

Explanation: The degree of a polynomial is the greatest exponent of the variable with a non-zero coefficient. Since \(r\neq0\), the coefficient of \(rx^7\) is non-zero. Hence the highest power of \(x\) is \(7\), so the degree is \(7\). Although \(3x^2\) has exponent \(2\), it is not the highest exponent. Exam tip: For a polynomial involving a parameter, always check whether the coefficient of the highest-power term is zero.

Correct answer: C. 2

Explanation: Putting r=0 makes rx^7=0. The polynomial therefore becomes 3x^2-10. Its highest power of x is 2, so its degree is 2. It is not 7 because the x^7 term has become zero. Exam tip: after substituting a parameter value, first remove all zero-coefficient terms and then identify the highest remaining exponent.

Correct answer: C. 5

Explanation: The powers of y in the polynomial are 5, 2, and 0; the constant term 8 has power 0. The greatest power is 5, so the degree of the polynomial is 5. The number 8 is a constant term, not the degree. Exam tip: To find the degree of a polynomial, identify the highest exponent of the variable.

Correct answer: C. 0

Explanation: To find the degree with respect to x, consider powers of x only. The expression 5y^2-3y+1 contains no x, so it is a non-zero constant polynomial with respect to x. Hence, its degree is 0. Option 2 is the highest power of y, not of x. Exam tip: Always check the powers of the variable specified in the question.

Correct answer: A. \(7x-4\)

Explanation: A linear polynomial has degree 1. In \(7x-4\), the highest power of \(x\) is 1, so it is a linear polynomial. \(x^2+3x-1\) has degree 2 and \(5x^3-2\) has degree 3, whereas \(9\) is a constant polynomial of degree 0. Exam tip: to find the degree of a non-zero polynomial, look for the highest exponent of the variable.

Correct answer: C. 6

Explanation: On expanding, \(2x^4(3x^2-1)=6x^6-2x^4\). The highest power of \(x\) is \(6\), so the degree of the polynomial is 6. Although \(-2x^4\) has degree 4, it is not the highest-degree term. Exam tip: simplify the expression first, then identify the highest power in a non-zero term.

Correct answer: C. 2

Explanation: On simplifying, \(x^2(x^3+1)-x^5+4x=x^5+x^2-x^5+4x=x^2+4x\). The terms \(x^5\) and \(-x^5\) cancel each other. The highest power of the variable in the remaining polynomial \(x^2+4x\) is 2, so its degree is 2. Choosing 5 would ignore the cancellation of like terms. Exam tip: Always simplify a polynomial completely before finding its degree.

Correct answer: B. The degree is the highest exponent of \(x\) in the polynomial, which is 5 here.

Explanation: The degree of a polynomial is the greatest exponent of the variable in any of its terms; the order of the terms does not matter. Here, \(-4x^5\) contains \(x\) with exponent 5, so the polynomial has degree 5. Option D is incorrect because a negative coefficient does not change an exponent. Exam tip: check every term and choose the largest exponent of the variable.

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

Explanation: In \(5x^3-2x+7\), the highest power of x is 3, so it is a degree 3 polynomial. Option B contains \(x^{-1}\), which is not allowed in a polynomial. Exam tip: powers must be non-negative integers.

Correct answer: C. 1

Explanation: Combining the like terms gives
\(x^4-2x^4+x^4=(1-2+1)x^4=0\). Hence, \(f(x)=3x-5\). The highest power of \(x\) in the simplified polynomial is 1, so its degree is 1. Option 4 may seem tempting from the original expression, but the fourth-degree terms cancel out. Exam tip: Always simplify like terms before finding the degree of a polynomial.

Correct answer: C. 5

Explanation: The total degree of a polynomial is the greatest sum of the exponents of variables in any one term. For 2x^3y^2, the sum is 3+2=5; for 5xy, it is 1+1=2; and the constant term -7 has degree 0. Hence, the highest total degree is 5. Note that 3 is only the exponent of x, not the total degree. Exam tip: In each term with two or more variables, add all exponents and select the greatest sum.

Correct answer: B. 5

Explanation: Find the total degree of each term: \(4x^2y^3\) has degree \(2+3=5\), \(-3xy^2\) has degree \(1+2=3\), and the constant term \(9\) has degree \(0\). The greatest of these is \(5\), so the total degree of the polynomial is \(5\). Option 3 is the degree of the second term, not of the whole polynomial. Exam tip: for a polynomial in several variables, add the exponents in each term and select the greatest sum.

Correct answer: B. 2

Explanation: To find the degree with respect to x, treat y as part of the coefficient. The powers of x in the terms are 2, 1, and 0 respectively. The greatest power is 2, so the degree of the polynomial is 2. The number 4 is the power of y, not the degree with respect to x. Exam tip: Look only at the highest exponent of the variable specified in the question.

Correct answer: C. 4

Explanation: To find the degree with respect to y, consider only the powers of y and treat x as a constant. The powers of y are 4 in 3x^2y^4, 1 in 7xy, and 0 in -1. Hence, the greatest power is 4, so the correct answer is 4. The value 6 comes from adding powers of x and y, which is not required when the degree is asked with respect to y. Exam tip: when a particular variable is specified, look only at its exponents.

Correct answer: C. (0)

Explanation: Terms with zero coefficients disappear and (5) remains. (5) is a non-zero constant polynomial of degree (0).

Correct answer: A. (t=1)

Explanation: For degree (2), the (x^3) term must vanish and the (x^2) term must remain. At (t=1), both conditions hold.

Correct answer: B. (4)

Explanation: To find the degree after assigning a value to a parameter, substitute that value first and simplify the polynomial. The degree is the greatest exponent of the variable whose coefficient is not zero. A term with a zero coefficient disappears completely, so it cannot determine the degree.

Here, with \(u=-2\), the coefficient of \(x^5\) is \(u+2=-2+2=0\), so that term vanishes. The coefficient of \(x^4\) is \(u^2-1=(-2)^2-1=4-1=3\), which is non-zero. The remaining polynomial is \(3x^4+x\). Its highest non-zero exponent is \(4\), so option B is correct. Option A would incorrectly retain the vanished \(x^5\) term.

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

Explanation: In \(7x^2-3x+5\), the highest power of \(x\) is 2, and all exponents are non-negative integers. Therefore, it is a polynomial of degree 2, i.e. a quadratic polynomial. \(x^3+2x-1\) has degree 3, so it is cubic. \(\frac{4}{x}+1\) contains \(x^{-1}\), and \(\sqrt{x}+2\) contains \(x^{1/2}\); hence, neither is a polynomial. Exam tip: a quadratic polynomial must have highest variable exponent exactly 2.

Correct answer: C. 3

Explanation: The expression is the product of three linear factors: x, (x-2), and (x+5). Each factor has degree 1, so the degree of the product is 1+1+1=3. On expansion, the highest-power term is x^3. Option 2 would be the degree of a product of only two linear factors. Exam tip: For a product of polynomials, add their degrees, provided the leading terms do not cancel.

Correct answer: B. \(4\)

Explanation: In the given expression, \(2x^6-2x^6=0\). Thus, \(g(x)=x^4-x^2\), and the highest power of \(x\) remaining is \(4\). Therefore, the degree of the polynomial is \(4\). It is not \(6\) because the \(x^6\) terms cancel each other. Exam tip: Always simplify like terms before finding the degree.

Correct answer: C. \(x+1\)

Explanation: In \(x+1\), the highest power of \(x\) is 1, so its degree is 1, not 0. The expressions \(8\), \(-3\), and \(\sqrt{2}\) are all non-zero constant polynomials; they contain no variable term, so each has degree 0. Exam tip: find the highest exponent of the variable to determine the degree of a polynomial.