Definition of a Polynomial

Correct answer: B. \(-7x^5+x^2-4\)

Explanation: The degree of a polynomial is the highest power of its variable, and the coefficient of that term is the leading coefficient. In \(-7x^5+x^2-4\), the highest power is 5 and the coefficient of \(x^5\) is \(-7\), which is negative. Hence, option B is correct. Option C has a negative leading coefficient, but its degree is only 4. Exam tip: identify the highest-power term first, then check its coefficient.

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

Explanation: First distribute \(x^2\) to each term inside the bracket: \(x^2(x^3-4)=x^5-4x^2\). The expression becomes \(x^5-4x^2-x^5+7x^2\). The like terms \(x^5-x^5=0\) cancel, and \(-4x^2+7x^2=3x^2\). Hence, the simplified expression is \(3x^2\). \(x^5+3x^2\) would result if the \(-x^5\) term were not combined. Exam tip: after expanding brackets, combine only like terms with the same variable and exponent.

Correct answer: B. Quadratic monomial

Explanation: The expression 3x^2 has only one term, so it is a monomial. The highest power of x is 2; therefore, its degree is 2 and it is a quadratic polynomial. Hence, it is a quadratic monomial. A linear polynomial has degree 1, while a constant polynomial has degree 0. Exam tip: To find the degree of a polynomial, identify the highest exponent of the variable.

Correct answer: B. (5)

Explanation: The degree of a non-zero polynomial is the greatest exponent of the variable whose coefficient is not zero. A term disappears completely when its coefficient becomes zero, so it must not be considered while finding the degree. This is especially important when a parameter is given a particular value, because that value may remove the term with the largest-looking exponent.

Put \(c=0\) into the expression. The coefficient of \(x^8\) becomes \(c=0\), so that term vanishes. The coefficient of \(x^5\) becomes \(c+2=2\), so the polynomial still contains the non-zero term \(2x^5\). The other terms have lower powers, namely \(-4x\) and \(1\). Therefore the greatest remaining exponent is \(5\), so option B is correct.

Correct answer: C. Cubic polynomial

Explanation: The degree of a polynomial is determined by the highest power of the variable having a non-zero coefficient. In (x^3+x+3), the highest power of x is 3, so it is a cubic polynomial. A quadratic polynomial has highest power 2. Exam tip: simplify the expression first and then identify its highest power.

Correct answer: C. 6

Explanation: This is a polynomial because all powers of \(x\) are non-negative integers. Since \(x^0=1\), the expression becomes \(x^6+x^4+1\). The highest power of \(x\) is 6, so its degree is 6. The number 4 is the degree of only one term, not of the whole polynomial. Exam tip: simplify terms such as \(x^0\) first, then identify the highest exponent.

Correct answer: B. 3

Explanation: The terms in the expression are \(8x^5\), \(0x^4\), \(-3x^2\), \(0x\), and \(6\). Any term with coefficient 0 is a zero term, so \(0x^4\) and \(0x\) are not counted. Only \(8x^5\), \(-3x^2\), and \(6\) are non-zero terms; therefore, the answer is 3. Choosing 4 would incorrectly include a term with zero coefficient. Exam tip: First remove all terms with coefficient 0 before counting polynomial terms.

Correct answer: B. When (u=-2)

Explanation: The direct answer is B, u=−2. A constant polynomial cannot contain x⁴ or x terms, so both their coefficients must be zero. Both coefficients are u+2. Thus set u+2=0, which gives u=−2. Substitution gives (−2+2)x⁴+(−2+2)x+15=0x⁴+0x+15=15, a non-zero constant polynomial. Option A, u=2, makes u+2=4, so variable terms remain. Option B is correct because it removes both variable terms at once. Option C, u=0, gives coefficient 2, so the expression is not constant. Option D, never, is wrong because u=−2 clearly works. The key point is that the same coefficient multiplies both variable terms, so one equation removes both. Memory cue: for a constant polynomial, set every coefficient of a positive power of x equal to zero.

Correct answer: B. \(0, 0, 0\)

Explanation: The polynomial is \(6x^8-5x^6+3\). It has no terms containing \(x^7\), \(x^5\), or \(x\), so the coefficient of each of these powers is \(0\). The numbers \(6\) and \(-5\) are coefficients of \(x^8\) and \(x^6\), respectively, so option A is not correct. Exam tip: If a power of a variable is absent from a polynomial, its coefficient is \(0\).

Correct answer: B. (4)

Explanation: To find the degree after a value is assigned to a parameter, substitute that value into every coefficient and then remove any term whose coefficient becomes zero. The degree is the largest exponent that remains attached to a non-zero term. A high exponent alone does not determine the degree if its coefficient vanishes.

For \(r=3\), the coefficient of \(x^7\) is \(r^2-9=3^2-9=9-9=0\), so the seventh-degree term disappears. The coefficient of \(x^4\) is \(r-1=3-1=2\), which is non-zero. The term \(x^2\) also remains. Thus the resulting polynomial has highest non-zero power \(4\), making option B the correct choice.

Correct answer: A. \(4x^3\)

Explanation: First distribute: \(x^3(x^2-5)=x^5-5x^3\). Thus, the expression becomes \(x^5-5x^3-x^5+9x^3\). Here \(x^5-x^5=0\) and \(-5x^3+9x^3=4x^3\), so the correct answer is \(4x^3\). \(x^5+4x^3\) is incorrect because the \(x^5\) terms cancel each other. Exam tip: after opening brackets, combine only like terms with the same power of the variable.

Correct answer: C. Cubic monomial polynomial

Explanation: The expression 4x^3 has only one term, so it is a monomial. The highest power of x is 3; therefore, its degree is 3 and it is a cubic polynomial. A quadratic polynomial has highest power 2, so option B is not correct. Exam tip: use the number of terms to identify monomial/binomial, and the highest exponent to identify the degree.

Correct answer: B. (2)

Explanation: The correct answer is B, degree 2. Substitute d=-1 into each coefficient: d+1=-1+1=0, so the x^5 term disappears. Also, 2d+3=2(-1)+3=-2+3=1, so the expression becomes 0x^5+1x^2+7=x^2+7. The highest exponent of x in a non-zero term is 2, so the degree is 2. Option A is wrong because the x^5 coefficient is zero; a zero term is not counted. Option B is correct. Option C is wrong because an x^2 term remains. Option D is wrong because the expression is not constant; it contains x^2. Exam cue: always substitute first, simplify zero coefficients, and then find the highest remaining exponent.

Correct answer: C. (a=4)

Explanation: For degree (3), the coefficient of (x^6) must be (0) and the coefficient of (x^3) must remain non-zero. From (2a-8=0), (a=4).

Correct answer: C. Because the variable is in the denominator

Explanation: In the original form, the denominator is (x-2), so it is not directly called a polynomial. Even after simplifying, the condition (x\neq2) remains.

Correct answer: D. Linear polynomial

Explanation: The degree of a polynomial is the highest power of its variable with a non-zero coefficient. In (x+2), the highest power of x is 1, so its degree is 1 and it is a linear polynomial. A quadratic polynomial must contain a term with x raised to the power 2, such as x². Exam tip: simplify the expression first, then identify the highest power of the variable.

Correct answer: B. \(x^2+\frac{1}{x}+5\)

Explanation: Option B contains \(\frac{1}{x}=x^{-1}\), so x has exponent \(-1\). In a polynomial in x, every exponent of x must be a non-negative integer. Options A, C, and D have only non-negative integer exponents, so they are polynomials. Exam tip: if a variable appears in the denominator or has a negative exponent, the expression is not a polynomial.

Correct answer: A. When (w=1)

Explanation: To become constant, both (w^2-1=0) and (w-1=0) are needed. Both conditions are satisfied together at (w=1).

Correct answer: A. (2^x+3)

Explanation: In (2^x), the variable is in the exponent. In a polynomial, the variable is the base with a non-negative integer power.

Correct answer: B. (6y-4)

Explanation: The degree of a polynomial is normally measured with respect to a specified variable. Here the variable is x, so an expression containing no x is treated as a constant in x. A non-zero constant polynomial has degree 0. This is different from the zero polynomial, whose degree is generally left undefined.

In option B, 6y-4 contains y but no x. If y is regarded as a fixed quantity while considering x, the whole expression is a non-zero constant with respect to x, so its degree in x is 0. Option C contains x and has degree 1, option D has x in a denominator and is not a polynomial, and option A is the zero polynomial. Thus B is the intended answer.

Correct answer: B. 0 and 0

Explanation: The expression x^4-4x^2+4 contains only the x^4 term, the x^2 term, and the constant term 4. It has neither an x^3 term nor an x term, so the coefficient of each missing power is taken as 0. Hence, the coefficients of x^3 and x are 0 and 0 respectively. The number -4 is the coefficient of x^2, not of x or x^3. Exam tip: Write a polynomial in descending powers of x and assign coefficient 0 to every missing power.

Correct answer: A. In \(x^5+x^2+1\), the coefficients of \(x^4\) and \(x^3\) are 0.

Explanation: \(x^5+x^2+1\) is a polynomial because all its exponents are non-negative integers. The \(x^4\) and \(x^3\) terms are absent, so their coefficients are taken as 0. Hence, it is a polynomial with missing powers. Option D is incorrect because the highest power is 5, so its degree is 5. Exam tip: Write 0 as the coefficient of every missing power.