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Chapter 5: Problem 8

Determine the coefficient of each term, the degree of each term, and the degree of the polynomial. $$12 x^{4} y-5 x^{3} y^{7}-x^{2}+4$$

Short Answer

Expert verified
The coefficients of the polynomial are 12, -5, -1, and 4. The degree of each term is 5, 10, 2 and 0 respectively. Therefore, the degree of the polynomial is 10.

Step by step solution

01

Determine Coefficients

The coefficients of the terms in the polynomial are the numbers in front of the variables: Thus, the coefficients of the polynomial \(12 x^{4} y -5 x^{3} y^{7} -x^{2} +4\) are 12 for the first term, -5 for the second term, -1 for the third term (since there's no number, we can consider the coefficient to be -1), and 4 for the fourth term.
02

Determine Degree of Each Term

To find the degree of each term, add the exponents of the variables in each term: For the first term \(12 x^{4} y\), the degree is \(4+1=5\). For the second term \(-5 x^{3} y^{7}\), the degree is \(3+7=10\). For the third term \(-x^{2}\), the degree is 2 (because there's only one variable, the degree is the exponent of the variable). The fourth term is a constant, so it's degree is 0.
03

Determine Degree of the Polynomial

The degree of the polynomial is the highest degree among all the terms in the polynomial. So by inspecting, the degree of the polynomial is 10 (which corresponds to the second term).

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