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Chapter 5: Problem 7

Determine the coefficient of each term, the degree of each term, and the degree of the polynomial. $$x^{3} y^{2}-5 x^{2} y^{7}+6 y^{2}-3$$

Short Answer

Expert verified
The coefficients of the terms in the polynomial are 1, -5, 6, and -3. The degrees of the terms in the polynomial are 5, 9, 2, and 0. The degree of the polynomial is 9.

Step by step solution

01

Identify the Coefficients

Examine each term of the polynomial \(x^{3}y^{2}-5x^{2}y^{7}+6y^{2}-3\) and identify its coefficient. The coefficient of the first term \(x^{3}y^{2}\) is 1 (it's assumed when a term doesn't have a specific coefficient displayed). The coefficient of the second term \(5x^{2}y^{7}\) is -5 (as indicated by the minus sign). The coefficient of the third term \(6y^{2}\) is 6 and the coefficient of the last term, which is a constant, is -3.
02

Identify the Degree of Each Term

The degree of a term in a polynomial is the sum of the exponents of the variables in that term. The degree of the first term \(x^{3}y^{2}\) is \(3+2 =5\), the degree of the second term \(5x^{2}y^{7}\) is \(2+7 = 9\), the degree of the third term \(6y^{2}\) is \(2\) because it has only one variable with exponent 2, and the degree of the last term, -3, is 0 because it has no variable.
03

Determine the Degree of the Polynomial

The degree of the polynomial is the highest degree of any term within the polynomial. Looking at the degrees identified in Step 2, the term with the highest degree is \(5x^{2}y^{7}\), which has a degree of 9. Thus, the degree of the polynomial is 9.

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