Degree of a Polynomial

The degree of a polynomial in one variable is the greatest exponent of that variable whose coefficient is not zero. A term with coefficient zero contributes nothing and must be ignored, even if its written exponent is larger than the exponents of the other terms. The condition \(t\neq0\) is essential because it confirms that \(tx^{11}\) is genuinely present and has a nonzero coefficient.

Here, \(0x^{12}\) is the zero term, so it cannot determine the degree. The remaining relevant powers are 11, 5, and 0, from \(tx^{11}\), \(3x^5\), and \(-1\). The greatest of these is 11. Therefore the polynomial has degree 11, making option B correct. Option A incorrectly counts the zero term.

In the given expression, the \(x^4\) and \(x^2\) terms cancel: \((x^4+x^2+1)-(x^4+x^2)=1\). Since \(1\) is a non-zero constant polynomial, its degree is \(0\). Degrees \(4\) and \(2\) relate to terms before simplification, not to the final polynomial. Exam tip: always simplify a polynomial completely before finding its degree.

The direct answer is option B, \(s=-1\). For degree 0, both variable terms must vanish, because only the constant 5 should remain. The coefficient of \(x^8\) is \(s^2-1\), and the coefficient of \(x^6\) is \(s+1\). At \(s=-1\), we get \((-1)^2-1=1-1=0\) and \(-1+1=0\). Hence the polynomial becomes 5, which has degree 0. Option B works exactly. Option A, \(s=1\), makes the \(x^8\) coefficient zero but gives the \(x^6\) coefficient 2, so the degree is 6. Option C, \(s=0\), gives coefficient -1 for \(x^8\), so the degree is 8. Option D, \(s=2\), leaves both higher-degree terms nonzero. A common mistake is checking only one coefficient; both must be zero.

On subtracting, the like terms cancel: \(A(x)-B(x)=(x^6+2x^3+1)-(x^6+2x^3)=1\). This is a non-zero constant polynomial, so its degree is \(0\). Option 6 is the degree of the original polynomials, but the \(x^6\) terms do not remain after subtraction. Exam tip: simplify a polynomial difference fully before identifying its degree.

\(A(x)+B(x)=(x^5+4)+(-x^5+3x^2)=3x^2+4\). The terms \(x^5\) and \(-x^5\) cancel each other. The highest exponent in the remaining polynomial is \(2\), so its degree is \(2\). Choosing \(5\) would be incorrect because no \(x^5\) term remains after addition. Exam tip: find the degree only after simplifying the resulting polynomial.

Using the identity \((a-b)(a+b)=a^2-b^2\), we get \((x^2-3)(x^2+3)=x^4-9\). Therefore, \(P(x)=x^4-9-x^4=-9\). Since \(-9\) is a non-zero constant polynomial, its degree is \(0\). The degree would be \(4\) only if the \(x^4\) terms did not cancel. Exam tip: simplify the polynomial first, then identify the highest power with a non-zero coefficient.

Substituting a=4 and b=9 gives the polynomial 2x^4+3x^9+1. The degree of a polynomial is the greatest exponent of x with a non-zero coefficient. Here, the greatest exponent is 9, so the degree is 9. Option 13 is incorrect because exponents of different terms are not added to find the degree. Exam tip: identify the exponent in each term and select the greatest one.

In option D, \(x^4-x^4=0\), so the expression simplifies to \(7x\). The highest exponent in \(7x\) is 1; hence its degree is 1, not 4. In option B, \(0x^5\) contributes nothing, but \(3x^4\) still makes the degree 4. Exam tip: simplify an expression and check for cancellation of like terms before finding its degree.

First expand: \(x^2(x^3+x)=x^5+x^3\). Therefore, \(p(x)=x^5+x^3-x^5-x^3+6=6\). A non-zero constant polynomial such as \(6\) has degree \(0\), so the correct answer is 0. Option 5 is tempting because it appears before simplification, but the highest-degree terms cancel. Exam tip: simplify a polynomial and combine like terms before finding its degree.

In x² - 3x + 5, the highest power of x is 2 and every exponent is a non-negative integer, so it is quadratic. Option B has degree 3. Exam tip: expressions with x in a denominator or under a root are not polynomials.

A polynomial of degree 0 is a non-zero constant polynomial. Therefore, the coefficients of both \(x^7\) and \(x^3\) must be zero. Since each coefficient is \(r-2\), we get \(r-2=0\), so \(r=2\). If \(r=-2\), the variable terms do not vanish. Exam tip: In degree-reduction questions, first make the coefficients of the highest-power terms zero.

The degree is the highest power of \(x\) having a non-zero coefficient. Thus, the \(x^4\) coefficient must be non-zero and no \(x^5\) or higher term can occur. Its coefficient need not be 1. Exam tip: identify the highest non-zero power.

The total degree of a polynomial is the greatest sum of the exponents of the variables in any one term. Here, the total degrees of x^3y^2, x^2y^4, and xy are 3+2=5, 2+4=6, and 1+1=2 respectively. Therefore, the highest total degree is 6. Option 5 is only the degree of the first term, not of the whole polynomial. Exam tip: Add the exponents in each term and select the greatest sum.

For the polynomial to have degree 0, only the constant term 3 must remain. Hence, the coefficients of both \(x^9\) and \(x^4\) must be zero. On substituting \(u=5\), \(u^2-25=25-25=0\) and \(u-5=0\), so the expression becomes 3. For \(u=-5\), the coefficient of \(x^9\) is zero, but the coefficient of \(x^4\) is \(-10\); therefore, its degree is 4. Exam tip: In degree-reduction questions, check the coefficients of every highest-power term that must vanish.

In the given expression, \(2x^8-2x^8=0\) and \(5x^4-5x^4=0\). Thus, \(F(x)=9x-1\). The highest power of \(x\) is \(1\), so the degree of the polynomial is \(1\). Degree \(0\) applies only to a non-zero constant polynomial. Exam tip: always simplify like terms before finding the degree.

((x^3-2)^2=x^6-4x^3+4), and after subtraction the higher variable terms cancel. The remaining (7x^2+4) has degree (2).

The direct answer is option B, \(a=-3\). A polynomial has degree 2 here because the terms in \(x^9\) and \(x^6\) must disappear, while the nonzero term \(4x^2\) must remain. For the \(x^9\) term, \(a+3=0\), which gives \(a=-3\). Check the \(x^6\) coefficient: \(a^2-9=(-3)^2-9=9-9=0\). The remaining polynomial is \(4x^2-5\), whose degree is 2. Option B is correct. Option A, \(a=3\), gives an \(x^9\) coefficient of 6, so the degree is 9. Option C, \(a=0\), gives coefficients 3 and -9, also leaving the degree 9 term. Option D, \(a=9\), leaves both high-degree terms. The important distinction is that degree 2 does not mean setting x equal to 2; it means all powers above 2 must vanish.

First expand: \(x^4(x^3+2)=x^7+2x^4\). Thus, the expression becomes \(x^7+2x^4-x^7+6x^5-8=6x^5+2x^4-8\). The \(x^7\) terms cancel, and the highest power of \(x\) in the remaining polynomial is 5, so its degree is 5. Option 7 is incorrect because the \(x^7\) term does not remain after simplification. Exam tip: expand brackets and combine like terms before finding a polynomial's degree.

On substituting \(k=4\), we get \(k-4=0\). Hence, both \((k-4)^2x^8\) and \((k-4)x^5\) become zero, leaving \(q(x)=3x\). The highest power of \(x\) in \(3x\) is \(1\), so the degree of the polynomial is \(1\). Although \(8\) and \(5\) occur in the original expression, their terms vanish because their coefficients become zero. Exam tip: substitute the parameter value first, then find the highest power among the remaining non-zero terms.

In option A, \(P+Q=(x^5+x^3)+(-x^5+2x^2)=x^3+2x^2\), which has degree 3. Leading terms may cancel. Exam tip: find the degree only after simplifying the sum.

In \(7x^4-3x+1\), the highest exponent of \(x\) is 4 and all exponents are non-negative integers, so it is a fourth-degree polynomial. \(x^4+\frac{1}{x}\) contains \(x^{-1}\), so it is not a polynomial. Exam tip: check every exponent.

For a polynomial in two variables, the total degree of a term is the sum of the exponents of all variables in that term. The term degrees are 3+4=7, 6+1=7, 1+2=3, and the constant has degree 0. The polynomial degree is the greatest term degree, so the answer is 7. Therefore, option B is correct; 6 counts only the highest power of x.

To find the degree with respect to x, treat y as part of the coefficients. The powers of x in the given expression are 2, 6, 0, and 0. The greatest of these is 6, so the degree is 6. The exponent 8 belongs to y, so it is not considered for the degree in x. Exam tip: Check the highest exponent only of the variable named in the question.

The degree of a polynomial with respect to \(y\) is the greatest exponent of \(y\). In the given expression, the exponents of \(y\) are \(8, 3, 2\), and \(0\). The greatest exponent is \(8\), so the correct answer is \(8\). Although \(x\) has exponent \(6\) in one term, it does not determine the degree with respect to \(y\). Exam tip: compare powers only of the variable specified in the question.

Since \(n>8\), the power \(n\) in the term \(4x^n\) is greater than 8. Thus, \(4x^n\) is the highest-degree term, and the degree of the polynomial is \(n\). Among the given choices, only \(n=9\) satisfies \(n>8\). Although \(n=8\) is the closest value, it does not satisfy the strict inequality. Exam tip: when finding degree, identify the greatest exponent and check every stated condition carefully.