Degree of a Polynomial

Correct answer: C. 0

On subtracting like terms, \(x^9-x^9=0\) and \(x^6-x^6=0\). Thus, the polynomial reduces to \(1\). Since \(1\) is a non-zero constant polynomial, its degree is \(0\). Degrees \(9\) and \(6\) occur in the original expression, but those terms cancel after simplification. Exam tip: Always simplify a polynomial completely before finding its degree.

Correct answer: B. (s=-13)

For degree (0), both (x^{14}) and (x^{10}) terms must vanish. At (s=-13), both coefficients become (0).

Correct answer: D. 0

On subtraction, the like terms cancel: \(A(x)-B(x)=(x^{11}+7x^6+17)-(x^{11}+7x^6)=17\). Since 17 is a non-zero constant polynomial, its degree is \(0\). Options 11 and 6 are degrees suggested by the original polynomials, not by their difference. Exam tip: always simplify a polynomial expression before identifying its highest power.

Correct answer: B. 7

\(A(x)+B(x)=(x^{10}+9)+(-x^{10}+11x^7)=11x^7+9\). The terms \(x^{10}\) and \(-x^{10}\) cancel each other. The highest exponent in the remaining polynomial is 7, so its degree is 7. Option 10 is incorrect because no term of degree 10 remains after addition. Exam tip: Always simplify the sum first, then identify the highest non-zero exponent.

Correct answer: A. \(\deg(PQ)=\deg P+\deg Q\)

When non-zero polynomials are multiplied, their leading terms multiply, so their exponents add: \(\deg(PQ)=\deg P+\deg Q\). In addition, leading terms may cancel, so option D is not always true. Exam tip: always check the non-zero condition before using the product-degree rule.

Correct answer: B. 15

Substituting a = 10 and b = 15 gives 6x^{10} + 8x^{15} + 1. The degrees of its terms are 10, 15, and 0. The highest exponent of a term with a non-zero coefficient is 15, so the degree of the polynomial is 15. Option 25 results from adding the exponents, but the degree of a polynomial is not found by adding exponents of separate terms. Exam tip: identify the exponent of x in every term and select the greatest one.

Correct answer: D. \(x^9-x^9+12x^6\)

In option D, \(x^9-x^9=0\), so the expression reduces to \(12x^6\). Its highest exponent is \(6\), so its degree is \(6\), not \(9\). In option B, the term \(0x^{10}\) vanishes, but \(8x^9\) still gives degree \(9\). Exam tip: simplify like terms and ignore terms with zero coefficients before finding the degree.

Correct answer: C. 0

First simplify the expression: \(x^7(x^3+x)=x^{10}+x^8\). Therefore, \(p(x)=x^{10}+x^8-x^{10}-x^8+59=59\). A non-zero constant polynomial such as \(59\) has degree \(0\), so the correct answer is \(0\). Choosing \(10\) or \(8\) is incorrect because those terms cancel out. Exam tip: Always simplify a polynomial completely before finding its degree.

Correct answer: C. 9

For a product of polynomials, the degrees are added. Here, \(x^7\) has degree 7 and \((x-6)^2\) has degree 2. Therefore, the degree of \(x^7(x-6)^2\) is \(7+2=9\). Option 14 is incorrect because the exponents are not multiplied. Exam tip: Add the degrees of polynomial factors in a product.

Correct answer: B. -12

A polynomial of degree 0 must be a non-zero constant, so the coefficients of all terms containing x must be zero. Here, the coefficient of both x^{12} and x^5 is r+12. Thus, r+12=0, giving r=-12. With this value, the polynomial becomes 26, whose degree is 0. Exam tip: A non-zero constant has degree 0, whereas the degree of the zero polynomial is not defined.

Correct answer: C. 17

For non-zero polynomials, the degree of a product equals the sum of their degrees. Hence, deg(p(x)q(x)) = 9 + 8 = 17, so 17 is correct. The value 9 is only the degree of p(x), not of the product. Exam tip: Add degrees when polynomials are multiplied; this rule does not always apply to addition or subtraction.

Correct answer: B. 15

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^8y^2, x^6y^9, and xy are 8+2=10, 6+9=15, and 1+1=2 respectively. The greatest value is 15, so the correct answer is 15. Option 10 is only the total degree of the first term, not of the whole polynomial. Exam tip: add the exponents in each term and select the largest sum.

Correct answer: C. 6

In the given expression, \(7x^{13}-7x^{13}=0\) and \(8x^9-8x^9=0\). Thus, \(F(x)=10x^6-5\) remains. Its highest exponent is \(6\), so the degree of the polynomial is \(6\). Options 13 and 9 are exponents of terms that cancel, so they cannot determine the degree. Exam tip: simplify by combining like terms first, then identify the highest exponent with a non-zero coefficient.

Correct answer: C. (8)

((x^8+5x^4)^2=x^{16}+10x^{12}+25x^8). After higher terms cancel, (25x^8+3x^7) remains, whose degree is (8).