Degree of a Polynomial

First expand the product: \((x^5+2x^2)(x^6-x^3)=x^{11}-x^8+2x^8-2x^5=x^{11}+x^8-2x^5\). Subtracting \(x^{11}\) leaves \(x^8-2x^5\). The highest power present is 8, so the degree is 8. Option 11 is a close distractor, but the \(x^{11}\) terms cancel. Exam tip: always simplify like terms before identifying the degree.

Direct answer: Option A, \\(c=6\\). A degree-zero polynomial is a nonzero constant, so every term containing a positive power of x must vanish. The coefficient of x^8 is c-6; setting it to zero gives c=6. The coefficient of x^5 is \\(c^2-36\\), and at c=6 it becomes \\(36-36=0\\) as well. The expression therefore reduces to the constant 31, whose degree is 0. Option A satisfies both cancellations. Option B, c=-6, makes the x^8 coefficient -12, so degree 8 remains, even though the x^5 coefficient becomes zero. Option C, c=0, leaves nonzero x^8 and x^5 terms. Option D, c=36, also leaves a nonzero x^8 coefficient. Do not confuse the value zero with degree zero: the remaining constant 31 must be nonzero.

The direct answer is option B, \(d=-4\). The degree is determined by the highest power of \(x\) whose coefficient is not zero. To make the degree 4, the \(x^7\) term must disappear, so \(d+4=0\), giving \(d=-4\). Now check the next term: \(d-2=-4-2=-6\), which is not zero, so the \(x^4\) term remains. The polynomial therefore becomes \(-6x^4+15\), of degree 4. Option A, \(d=2\), removes the \(x^4\) term but leaves \(6x^7\), so its degree is 7. Option B removes only the higher-degree term and keeps the fourth-degree term. Option C gives a non-zero \(x^7\) coefficient, so the degree is 7. Option D also leaves the seventh-degree term. Exam cue: first cancel the unwanted highest power, then verify that the required lower power remains.

To find the degree with respect to \(z\), look at the powers of \(z\) in each term: \(7x^4z^8\) has \(z^8\), \(-6x^2z^6\) has \(z^6\), and \(5x^3-4\) has \(z^0\) (no \(z\) factor). The highest power of \(z\) is \(8\). So the degree with respect to \(z\) is \(8\).

The polynomial \(4x^7-3\) has only two terms, but its highest exponent of \(x\) is 7, so its degree is 7, not 2. Exam tip: count the highest power, not the terms.

This product has the form

a^2-b^2=(a-b)(a+b)

. Taking

a=x^6

and

b=1

gives

(x^6-1)(x^6+1)=x^{12}-1

. The highest power of

x

in this polynomial is 12, so its degree is 12. Degree 6 is the degree of each factor, not of the complete product. Exam tip: Recognising an identity before multiplying helps find the degree quickly.

The leading coefficients of non-zero polynomials are non-zero, so the leading term of their product cannot cancel. Hence, the degree of a product is the sum of degrees. In a sum, leading terms may cancel; remember this exception in exams.

On removing the brackets, every term of the second expression changes sign: \(p(x)=x^7-5x^4+2-x^7+2x^4=-3x^4+2\). The \(x^7\) terms cancel, and the highest power in the remaining polynomial is \(4\); hence its degree is \(4\). Although \(7\) appears in the original expressions, its term does not remain after simplification. Exam tip: always simplify a polynomial completely before finding its degree.

On expanding the expression,
\(x^7(x^2+1)-x^2(x^7-6)=x^9+x^7-x^9+6x^2=x^7+6x^2\). The highest power of \(x\) in the simplified polynomial is 7, so its degree is 7. Option 9 is incorrect because the \(x^9\) terms cancel each other. Exam tip: Always expand and combine like terms before finding the degree of a polynomial.

On simplification, every term has coefficient 0, so p(x)=0. This is the zero polynomial, and the degree of the zero polynomial is not defined. Choosing 9 is incorrect because the coefficient of x^9 is also 0, so that term does not contribute to the polynomial. Exam tip: Remove terms with zero coefficients before finding the degree.

The leading \(x^n\) terms are added. If their coefficients sum to 0, the \(x^n\) term cancels, so the degree may decrease. Equal constant terms do not affect degree. Exam tip: check the next non-zero term.

In \(x^6y^3+2x\), the term \(x^6y^3\) has total degree \(6+3=9\), while \(2x\) has degree 1. Therefore, the total degree of the polynomial, which is the greatest total degree among its terms, is 9. In option A, the greatest total degree is only \(4+3=7\). Exam tip: For a term in two variables, add the exponents of all variables to find its total degree.

The degree of a polynomial is found from the greatest exponent with a nonzero coefficient. A term such as \(0x^{17}\) is identically zero and does not affect the polynomial. The condition \(t\neq0\) confirms that \(tx^{15}\) remains present and must be compared with the lower-power terms.

Once the zero term is ignored, the exponents present are 15, 8, and 0. The largest of these is 15, and its coefficient t is nonzero by assumption. Therefore the degree is 15, so option B is correct. The tempting answer 17 is wrong because the coefficient of \(x^{17}\) is zero; a formal exponent alone cannot determine polynomial degree.

On expanding the brackets, the like terms cancel:
\((x^7+x^4+1)-(x^7+x^4)=1\). This is a non-zero constant polynomial. The degree of every non-zero constant polynomial is \(0\), so the correct answer is 0. “Not defined” applies to the zero polynomial, not to the constant polynomial 1. Exam tip: simplify the expression first, then identify the highest power that remains.

For degree (0), both (x^{12}) and (x^8) terms must vanish. At (s=-5), both coefficients become (0).

On subtraction, the like terms cancel: \(A(x)-B(x)=(x^9+5x^4+11)-(x^9+5x^4)=11\). Since 11 is a non-zero constant polynomial, its degree is \(0\). Options 9 and 4 refer to powers in the original polynomials, but those terms do not remain after subtraction. Exam tip: always simplify a sum or difference of polynomials before finding its degree.

\(A(x)+B(x)=(x^8+7)+(-x^8+9x^5)=9x^5+7\). The \(x^8\) terms cancel each other. The highest power with a non-zero coefficient in the remaining polynomial is \(5\), so its degree is \(5\). Choosing \(8\) would be incorrect because no \(x^8\) term remains after addition. Exam tip: always simplify the sum first, then identify the highest power with a non-zero coefficient.

Combining like terms gives \(3x^6-3x^6=0\) and \(-2x^4+5x^4=3x^4\). Thus, \(R(x)=3x^4+7x^2\), whose highest exponent is 4. In exams, always simplify before finding degree.

On substituting a=8 and b=13, the polynomial becomes 4x^8+6x^13+1. The degrees of its terms are 8, 13, and 0. Since 13 is the highest exponent of x, the degree of the polynomial is 13. Note that 6 is only the coefficient of x^13, not the degree. Exam tip: identify the largest exponent of the variable after simplifying or substituting values.

First simplify the expression: \(x^5(x^3+x)=x^8+x^6\). Therefore, \(p(x)=x^8+x^6-x^8-x^6+37=37\). Since 37 is a non-zero constant polynomial, its degree is \(0\). The values \(8\) and \(6\) are powers of terms before simplification, but those terms cancel. Exam tip: always simplify a polynomial completely before finding its degree.

The degree of \(x^5\) is 5, and the degree of \((x-4)^2\) is 2. The degree of the product of two non-zero polynomials equals the sum of their degrees. Therefore, the degree is \(5+2=7\). Option 10 is incorrect because the degrees are not multiplied. Exam tip: In a product, add the degrees; for a power of a factor, multiply its degree by the power.

For the polynomial to have degree 0, it must become a constant polynomial. Therefore, the coefficients of both \(x^{10}\) and \(x^3\) must be zero. Their common coefficient is \(r+9\), so \(r+9=0\), giving \(r=-9\). The expression then becomes only \(18\), whose degree is 0. If \(r=0\), the coefficient of \(x^{10}\) is 9, so the degree remains 10. Exam tip: first check the coefficient of the term with the highest power.

For the product of two non-zero polynomials, the degree equals the sum of their degrees. Thus, \(\deg(p(x)q(x))=7+6=13\), so 13 is correct. Option 42 comes from multiplying the degrees, which is not the rule for the degree of a polynomial product. Exam tip: add the degrees when polynomials are multiplied.

Find the total degree of each term: \(x^6y^2\) has degree \(6+2=8\), \(x^4y^7\) has degree \(4+7=11\), and \(xy\) has degree \(1+1=2\). The greatest of these is 11, so the total degree of the polynomial is 11. Option 8 is only the degree of the first term, not of the whole polynomial. Exam tip: For a polynomial in more than one variable, add the exponents in each term and select the greatest sum.

On combining like terms, \(5x^{11}-5x^{11}=0\) and \(6x^7-6x^7=0\). Thus, \(F(x)=8x^4-3\). The highest exponent of \(x\) is 4, so the degree of the polynomial is 4. Option 0 would be the degree of a non-zero constant polynomial, but an \(x^4\) term remains here. Exam tip: simplify and cancel like terms before finding a polynomial's degree.