Degree of a Polynomial

In \(5x^3-2x+7\), the powers of \(x\) are 3, 1, and 0, so the highest power is 3. Hence it is a cubic polynomial. Option B has \(1/x=x^{-1}\), so it is not a polynomial. Exam tip: check for non-negative integer powers.

The total degree of a polynomial in more than one variable is the greatest sum of the exponents of the variables in any term. Here, the term degrees are 7+4=11, 3+10=13, 1+5=6, and 0 for the constant term -13. The greatest value is 13, so the total degree is 13. Although 11 is the degree of the first term, it is not the highest. Exam tip: Add the exponents in each term and choose the largest sum.

To find the degree with respect to x, treat y as part of the coefficients. The powers of x in the given expression are 9, 7, 0, and 0. The greatest of these is 9, so the degree is 9. The exponent 11 belongs to y and is not considered for the degree in x. Exam tip: Look only at the highest exponent of the variable specified in the question.

The degree is the greatest exponent of the variable in any term. Here the exponents are 3, 1 and 0, so the degree is 3. Adding exponents is incorrect. Exam tip: first identify the term with the highest power and a non-zero coefficient.

The degree of a polynomial is the greatest exponent of its variable among its terms. The term \(3x^{11}\) makes the degree \(11\). Since \(m<11\), \(m=9\) is possible, and the exponent of \(8x^9\) remains below \(11\). \(m=11\) does not satisfy \(m<11\), while \(m=12\) or \(m=13\) would make the degree greater than \(11\). Exam tip: Check both the highest exponent and every condition given in the question.

The \(x^5\) terms cancel because their coefficients are opposites: \(6x^5-6x^5=0\). The simplified polynomial is \(4x^2-1\), whose highest exponent is 2. Exam tip: always simplify before finding degree.

The zero polynomial \(0\) has no non-zero term, so there is no highest exponent to identify; hence its degree is undefined. In contrast, \(-7\) is a non-zero constant of degree 0. Exam tip: only the zero polynomial has undefined degree.

The term with the greatest possible exponent is \\( (b+8)x^{16}\\). The condition \\(b\\ne-8\\) means that \\(b+8\\) is nonzero. Hence the degree-16 term is present and cannot be cancelled by any other displayed term, since the other powers are only 7 and 0. The polynomial therefore has degree 16, so option A is correct.

The term \\(10x^7\\) has degree 7 and the constant \\(-3\\) has degree 0. These lower-degree terms do not replace or reduce the degree of a nonzero degree-16 term. Only if \\(b=-8\\) would the leading term disappear and the degree become 7. That excluded value explains why the answer is definitely 16.

Substituting b=-8 gives b+8=0. Hence, the x^{16} term becomes 0x^{16} and vanishes, leaving the polynomial 10x^7-3. The highest power of x is 7, so its degree is 7. Option 16 is incorrect because its coefficient becomes zero. Exam tip: Before finding a polynomial’s degree, check whether the coefficient of the apparent highest-degree term becomes zero.

First simplify: \(P(x)=x^4+2x^2+1-x^4+3x^2-5=5x^2-4\). The highest power with a non-zero coefficient is 2, so the degree is 2. Exam tip: combine like terms before finding degree.

First expand: \(7x^9(x^2-1)=7x^{11}-7x^9\). Thus, the expression becomes \(7x^{11}-7x^9-7x^{11}+6x^8=-7x^9+6x^8\). The \(x^{11}\) terms cancel, and the highest remaining exponent is 9, so the degree is 9. Option 8 is the exponent of the other remaining term, not the highest one. Exam tip: Always simplify a polynomial completely before finding its degree.

Terms with zero coefficients disappear and (-61) remains. It is a non-zero constant polynomial with degree (0).

Here, \(Y(x)=0\) is the zero polynomial, meaning that all its coefficients are zero. It has no non-zero term with a highest power, so its degree is not defined. \(0\) is the degree of a non-zero constant polynomial, such as \(5\), not of the zero polynomial. Exam tip: To find a degree, look for the highest power having a non-zero coefficient.

First simplify the expression: \(x^7(x^2-8)=x^9-8x^7\). Hence, \(p(x)=x^9-8x^7-x^9+8x^7+43=43\). This is a non-zero constant polynomial, so its degree is \(0\). Choosing 9 would ignore the cancellation of the \(x^9\) terms. Exam tip: always simplify a polynomial completely before finding its degree.

For degree (0), both (x^{10}) and (x^7) terms must vanish. At (c=8), both coefficients become (0).

The direct answer is option B, \(d=-9\). A degree of 6 means the ninth-degree term must vanish, but the sixth-degree term must remain. The coefficient of \(x^9\) is \(d+9\), so set \(d+9=0\), giving \(d=-9\). The coefficient of \(x^6\) then is \(d-4=-9-4=-13\), which is non-zero. Thus the polynomial becomes \(-13x^6+19\), and its degree is 6. Option A, \(d=4\), removes the \(x^6\) term but leaves \(13x^9\), giving degree 9. Option B gives exactly the required condition. Option C leaves a non-zero ninth-degree coefficient. Option D also leaves the ninth-degree term. Exam cue: for a target degree, cancel every term above it and check that the target coefficient is not zero.

The phrase “with respect to \(z\)” tells us which variable is being treated as the polynomial variable. The degree is the greatest exponent of \(z\) having a non-zero coefficient. Expressions involving other variables, such as \(x^6\), are treated as coefficients when the degree is considered with respect to \(z\).

In \(9x^6z^{10}-8x^2z^7+5x^3-6\), the powers of \(z\) are 10, 7, 0, and 0. The coefficient of \(z^{10}\) is \(9x^6\), which is non-zero as a polynomial coefficient, so the highest relevant exponent is 10. Therefore option C is correct. The exponent 6 belongs to \(x\), not \(z\), and the constant terms have degree zero in \(z\).

Like terms must be combined first: \(5x^3-2x^3=3x^3\). Thus, \(p(x)=3x^3+x^2-7\), whose highest exponent is 3. Exponents are added during multiplication, not during addition or subtraction. Exam tip: simplify the polynomial before identifying its degree.

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On opening the brackets, \(p(x)=x^3+2x^2-x-x^3+5x-4=2x^2+4x-4\). The \(x^3\) terms cancel, so the highest remaining power is 2. Exam tip: simplify fully before finding degree.

On opening the brackets, \(p(x)=x^9-7x^6+5-x^9+3x^6=-4x^6+5\). The \(x^9\) terms cancel, and the highest exponent in the remaining polynomial is \(6\), so its degree is \(6\). The number \(5\) is a constant term, not the degree. Exam tip: When subtracting a bracket, change the sign of every term before simplifying.

Expanding the expression gives \(x^9(x^2+1)-x^2(x^9-8)=x^{11}+x^9-x^{11}+8x^2=x^9+8x^2\). The \(x^{11}\) terms cancel each other. The highest power of \(x\) in the remaining polynomial is \(9\), so its degree is \(9\). Choosing \(11\) would be incorrect because that term does not remain after simplification. Exam tip: always simplify a polynomial fully before finding its degree.

Every term in the given expression has coefficient \(0\), so \(p(x)\) is the zero polynomial. The zero polynomial has no non-zero term, so it has no highest power and its degree is not defined. \(0\) is the degree of a non-zero constant polynomial, not of the zero polynomial. Exam tip: Before finding a degree, check whether the polynomial is the zero polynomial.

The degree is the greatest exponent of the variable among terms with non-zero coefficients, not the largest coefficient. Since \(2x^6\) has exponent 6, the degree is 6. Exam tip: scan exponents first, not coefficients.

In \(x^8y^3+2x\), the term \(x^8y^3\) has total degree \(8+3=11\). The other term, \(2x\), has degree 1, so the greatest total degree of the polynomial is 11. Hence, option B is correct. Option A has degree \(6+4=10\). Exam tip: For a term in several variables, add the exponents of all its variables to find its total degree.