Degree of a Polynomial

The degree of a polynomial is the greatest exponent of its variable. The term \(4x^6\) makes the degree 6. Since \(m<6\), \(m=4\) is possible, and the exponent in \(3x^4\) remains less than 6. Although \(m=6\) would not change the highest exponent, it does not satisfy \(m<6\); \(m=7\) and \(m=8\) would make the degree 7 or 8. Exam tip: In degree questions, first identify the highest exponent and then check every given condition.

Since \(0x^6\) is a zero term, it does not affect the degree. The highest power among the non-zero terms is \(x^4\), so the degree is 4. Exam tip: ignore all terms with coefficient zero first.

Since \(7\) has no variable, it is a constant polynomial with degree 0. \(7x\) and \(x+7\) contain the variable, while \(\frac{7}{x}=7x^{-1}\) is not a polynomial. Exam tip: powers of a variable in a polynomial cannot be negative.

The governing concept is that the degree is determined by the highest power whose coefficient is non-zero. In this polynomial, the coefficient of x^10 is a - 1. The condition a ≠ 1 guarantees that a - 1 ≠ 0, so the x^10 term is present and cannot disappear. Since x^10 has a higher power than x^4 and the constant term has power 0, the degree of the polynomial is 10. Therefore option C is correct. Option A selects the degree of the lower non-constant term and would be relevant only if the x^10 coefficient vanished. Option B is not an exponent present in the expression, and option D confuses the degree with a coefficient or with the difference a - 1. The condition on a is precisely what preserves the leading term.

Substituting a=1 gives a-1=0, so the term 0x^{10} vanishes. The polynomial becomes 7x^4-8, whose highest power of x is 4. Hence, its degree is 4. Option 0 would apply only if no variable term remained. Exam tip: first substitute the parameter value, then identify the highest non-zero power of the variable.

In the zero polynomial, the coefficient of every power is 0, so there is no non-zero term with a highest power. Therefore, under the standard Class 9 definition, its degree is not defined. Option 0 is the degree of a non-zero constant polynomial, not of the zero polynomial. Exam tip: while finding degree, consider only terms with non-zero coefficients.

First expand: \(2x^4(x^3-1)=2x^7-2x^4\). Thus, the expression becomes \(2x^7-2x^4-2x^7+9x^5=9x^5-2x^4\). The highest power with a non-zero coefficient is \(5\), so the degree of the polynomial is 5. Option 7 is incorrect because \(2x^7\) and \(-2x^7\) cancel each other. Exam tip: Expand brackets and combine like terms before finding the degree.

Terms with zero coefficients disappear and (-17) remains. It is a non-zero constant polynomial of degree (0).

The degree of a polynomial is its greatest exponent with a non-zero coefficient. The coefficient of a missing term is understood to be zero. Thus we must check both requirements: the highest present power must be 7, and the x^6 term must be absent. A term such as x^5 does not affect either requirement, provided the x^7 term remains present.

Option A is 3x^7-2x^5+1. Its x^7 coefficient is 3, so its degree is 7. There is no x^6 term, meaning its coefficient is 0. Option B contains x^6, option C has degree 6 rather than 7, and option D has degree 8. Therefore option A satisfies both conditions.

This is the zero polynomial because all its coefficients are 0. The degree of a polynomial is determined by the highest power having a non-zero coefficient. The zero polynomial has no such term, so its degree is not defined. Degree 0 applies to a non-zero constant polynomial, not to the zero polynomial. Exam tip: Always distinguish the zero polynomial from a non-zero constant polynomial.

First simplify the expression: \(x^2(x^2-1)=x^4-x^2\). Therefore, \(p(x)=x^4-x^2-x^4+x^2+5=5\). This is a non-zero constant polynomial, so its degree is \(0\). Option 4 results from noticing the highest power before simplification, but those terms cancel out. Exam tip: always simplify a polynomial completely before finding its degree.

Expanding, \((x^2+2x)(x^3-x)=x^5+2x^4-x^3-2x^2\). On subtracting \(x^5\), the \(x^5\) terms cancel, leaving \(2x^4-x^3-2x^2\). Its highest exponent is \(4\), so the degree is \(4\). Option \(5\) is not correct because the \(x^5\) term has been cancelled. Exam tip: determine the degree only after simplifying and combining like terms.

The direct answer is option B, \(c=-2\). A polynomial has degree 0 when it is a nonzero constant, so every term containing x must disappear. The coefficient of \(x^5\) is \(c+2\). Setting it to zero gives \(c+2=0\), hence \(c=-2\). Check the other coefficient: \(c^2-4=(-2)^2-4=4-4=0\). The expression therefore becomes only \(9\), whose degree is 0. Option B is correct. Option A, \(c=2\), makes the first coefficient 4, so an \(x^5\) term remains. Option C, \(c=0\), gives coefficients 2 and -4, so the degree is still 5. Option D, \(c=4\), also leaves both variable terms present. The safe method is to make the coefficients of all positive powers zero and then check that the constant term is not zero.

The governing concept is that a polynomial has degree 2 when its x^4 coefficient is zero while its x^2 coefficient remains non-zero. The coefficient of x^4 is d - 1, so the x^4 term disappears when d - 1 = 0, giving d = 1. At this value, the coefficient of x^2 is d + 3 = 1 + 3 = 4, which is non-zero. The expression therefore becomes 4x^2 + 6 and has degree 2. Hence option A is correct. If d = -3, the x^2 term vanishes but the x^4 coefficient is -4, so the degree is 4. For d = 0 or d = 4, the x^4 coefficient is also non-zero, so the degree remains 4. Both conditions must be checked, not just the cancellation of one term.

For a polynomial in several variables, the total degree is the greatest sum of exponents in any one term. The term \(x^2y^3z^4\) has total degree \(2+3+4=9\). The term \(3xy^2z\) has total degree \(1+2+1=4\), while the constant \(-5\) has degree 0. Hence, the greatest total degree is 9. Exam tip: add the exponents within each term and choose the largest sum; do not multiply the exponents.

The phrase “with respect to \(z\)” means that only the powers of \(z\) are used to determine the degree. The other symbols, such as \(x\), are treated as coefficients while viewing the expression as a polynomial in \(z\). Therefore powers of \(x\) must not be added to powers of \(z\), and the total degree in all variables is a different question.

Inspect the terms according to their powers of \(z\): \(4x^2z^5\) has \(z\)-power \(5\), \(-3xz^3\) has power \(3\), \(8x\) has power \(0\), and \(-1\) also has power \(0\). The largest of these is \(5\), with a non-zero coefficient \(4x^2\). Hence the degree with respect to \(z\) is \(5\), so option C is correct.

We have \((x^2+x)^2=x^4+2x^3+x^2\). Therefore, \(h(x)=x^4+2x^3+x^2-x^4-2x^3=x^2\). The degree of a non-zero polynomial is its highest power of the variable, so the degree is \(2\). Choosing \(4\) or \(3\) would ignore the cancellation of like terms. Exam tip: always simplify a polynomial fully before finding its degree.

For degree 5, the coefficient of \(x^7\) must become zero. Thus, \(a-5=0\), so \(a=5\). The highest remaining non-zero power is then \(x^5\). Exam tip: always check whether the leading coefficient can vanish.

The degree of a polynomial is its greatest exponent with a non-zero coefficient. Here the exponents are \(n+2\), \(5\), and \(0\). For the degree to remain \(5\), \(n+2\) must not exceed \(5\); hence \(n+2\leq 5\), or \(n\leq 3\). Although \(n=0\) is one possible value, it is not the complete condition. Exam tip: In degree questions, compare the exponents of all terms and identify the greatest one.

Simplifying the expression gives
\(p(x)=x^4-2x^2+1-x^4+x^2=-x^2+1\). The \(x^4\) terms cancel, and the highest power of \(x\) in the remaining polynomial is 2. Therefore, its degree is 2. Option 4 is incorrect because the \(x^4\) term does not remain after simplification. Exam tip: Always simplify a polynomial completely before finding its degree.

On expanding, \(x^4(x^2+1)-x^2(x^4-3)=x^6+x^4-x^6+3x^2=x^4+3x^2\). The \(x^6\) terms cancel each other. The highest power in the remaining polynomial is \(4\), so its degree is 4. Choosing \(6\) would be an error caused by looking at the powers before simplification. Exam tip: Always simplify a polynomial completely before finding its degree.

A parameter multiplying the highest power may make that term disappear. Therefore, one must substitute the given value and inspect the remaining terms before stating the degree. The degree is the greatest exponent with a non-zero coefficient, not necessarily the greatest exponent written in the original expression.

For \(e=0\), the coefficient of \(x^9\) is zero, so \(ex^9=0\). The coefficient of \(x^7\) is \(e-2=0-2=-2\), which is non-zero. The expression becomes \(-2x^7+5x^2-1\). Its highest non-zero power is 7, so option B is correct. The term \(5x^2\) and the constant do not change the answer because their powers are lower. Option A incorrectly counts the vanished ninth-degree term.

The expression \(0x^5+0x^3+0x+0\) simplifies to \(0\), so it is the zero polynomial. The zero polynomial has no non-zero term with a highest power; therefore, its degree is not defined. Choosing \(5\) is incorrect because the coefficient of \(x^5\) is \(0\). Exam tip: Before finding degree, remove all terms whose coefficients are \(0\).

For \(a=0\), the term \(ax^5\) vanishes, leaving \(3x^4-2\), whose highest power is 4. For every non-zero \(a\), the \(x^5\) term remains. Exam tip: remove zero-coefficient terms before identifying degree.

In \(x^3y^3+2x\), the term \(x^3y^3\) has total degree \(3+3=6\). The total degree of a polynomial is the greatest total degree among its terms, so this polynomial has total degree 6. In option A, the greatest total degree is only 4, so it is not correct. Exam tip: Add the exponents of all variables in each term and select the greatest sum.