Problem 61 - Algebra and Trigonometry

Question

In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. $$ \left(4 x^{2} y+8 x y+11\right)+\left(-2 x^{2} y+5 x y+2\right) $$

Step-by-Step Solution

Answer

The result of the given operation is the polynomial \(2x^{2}y + 13xy + 13\) and the degree of this polynomial is 3.

Step 1: Identify like terms

In the given problem, the expression can be rewritten as \(4x^{2}y+8xy+11-2x^{2}y+5xy+2\). The like terms in this expression are \(4x^{2}y\) and \(-2x^{2}y\), \(8xy\) and \(5xy\), and the constants \(11\) and \(2\).

Step 2: Add the like terms

Next, add the coefficients together for each pair of like terms. The term \(4x^{2}y - 2x^{2}y\) becomes \(2x^{2}y\). The term \(8xy + 5xy\) results in \(13xy\). And finally, add the constants \(11 + 2\) to get \(13\). This results in a new polynomial \(2x^{2}y + 13xy + 13\).

Step 3: Determine the degree of the polynomial

The degree of a polynomial is the highest exponent of the variable. In this scenario, looking at each term in the polynomial \(2x^{2}y + 13xy + 13\), the term with the highest exponent is \(2x^{2}y\). In this term, the exponent of \(x\) is 2 and \(y\) is 1. Therefore, the degree of the polynomial is 2+1=3.

Key Concepts

Like Terms

When working with polynomials, identifying like terms is crucial. Like terms are terms that have the exact same variable parts, which means they must have the same variable(s) raised to the same power. For instance, consider the expression \(4x^{2}y + 8xy + 11 - 2x^{2}y + 5xy + 2\). Here, the like terms are:

• \(4x^{2}y\) and \(-2x^{2}y\)
• \(8xy\) and \(5xy\)
• The constants \(11\) and \(2\)

Recognizing these allows us to combine them effectively in polynomial operations, making it simpler to solve or simplify an expression.

Degree of a Polynomial

The degree of a polynomial is one of its most important properties. It is defined as the highest sum of the exponents of the variables in a monomial (single term) of the polynomial. For example, in the polynomial \(2x^{2}y + 13xy + 13\), you would assess each term:

• \(2x^{2}y\): the powers add up to \(2 + 1 = 3\)
• \(13xy\): the powers add up to \(1 + 1 = 2\)
• \(13\): no variables, hence degree \(0\)

The term \(2x^{2}y\) determines the degree of the polynomial, which is \(3\). Hence, the degree of this polynomial is \(3\). Knowing the degree of a polynomial is crucial, especially when determining the behavior and the roots of the polynomial function.

Polynomial Addition

Adding polynomials involves combining like terms to simplify the expression. Look at the example given: \(4x^{2}y + 8xy + 11 + (-2x^{2}y + 5xy + 2)\). You add the coefficients of like terms:

• For the terms \(4x^{2}y\) and \(-2x^{2}y\), you get \(2x^{2}y\)
• For the terms \(8xy\) and \(5xy\), it results in \(13xy\)
• For the constants \(11\) and \(2\), you sum them to \(13\)

Thus, polynomial addition simplifies the expression to \(2x^{2}y + 13xy + 13\). This process helps in reducing complex polynomial expressions into more manageable and useful forms.

Coefficients

In any polynomial, coefficients are the numerical values that multiply the variables or the terms. For example, in the polynomial \(2x^{2}y + 13xy + 13\):

• The coefficient of \(x^{2}y\) is \(2\)
• The coefficient of \(xy\) is \(13\)
• The constant \(13\) can be considered a term with an implicit variable raised to the zero power, with \(13\) as its coefficient

Understanding coefficients is helpful because they tell you how much each term contributes to the overall value of the polynomial. They are essential in performing operations, such as addition and subtraction, and determining the polynomial's growth rate in functions.