Problem 83 - Beginning and Intermediate Algebra

Question

Each of the polynomials below is a polynomial in two variables. Perform the indicated operation(s). $$\begin{array}{l} \left(8 x^{3} y^{2}-7 x^{2} y^{2}+7 x^{2} y-3\right)+\left(2 x^{3} y^{2}+x^{2} y-1\right) \\ -\left(4 x^{2} y^{2}+2 x^{2} y+8\right) \end{array}$$

Step-by-Step Solution

Answer

The simplified polynomial after performing the indicated operations (addition and subtraction) is: \(P = 10x^3y^2 - 9x^2y^2 + 6x^2y - 12\)

Step 1: Identify Like Terms in Each Polynomial

Let's first rewrite the given exercise to more clearly see the indicated operations and to identify any like terms in the polynomials. The polynomials are: \(P1 = 8x^3y^2 - 7x^2y^2 + 7x^2y - 3\) \\ \(P2 = 2x^3y^2 + x^2y - 1\) \\ \(P3 = 4x^2y^2 + 2x^2y + 8\)

Step 2: Perform the Indicated Operations

Now, perform the indicated operations by adding P1 and P2 and then subtracting P3: \(P = (P1 + P2) - P3\)

Step 3: Simplify the Addition of P1 and P2

To add P1 and P2, combine like terms: \((P1 + P2) = (8x^3y^2 - 7x^2y^2 + 7x^2y - 3) + (2x^3y^2 + x^2y - 1)\) \(P1 + P2 = 10x^3y^2 - 7x^2y^2 + 2x^2y^2 + 7x^2y + x^2y - 3 - 1\)

Step 4: Perform the Subtraction of P3

Now, subtract P3 from the result of P1 and P2: \(P = (P1 + P2) - P3\) \(P = (10x^3y^2 - 7x^2y^2 + 2x^2y^2 + 7x^2y + x^2y - 3 - 1) - (4x^2y^2 + 2x^2y + 8) \)

Step 5: Simplify by Combining Like Terms

Simplify the expression by combining like terms: \(P = 10x^3y^2 - 7x^2y^2 + 2x^2y^2 - 4x^2y^2 + 7x^2y + x^2y - 2x^2y - 3 - 1 - 8\) \(P = 10x^3y^2 - 9x^2y^2 + 6x^2y - 12\) So the final simplified polynomial is: \(P = 10x^3y^2 - 9x^2y^2 + 6x^2y - 12\)

Key Concepts

Like Terms

When working with polynomials, understanding **like terms** is crucial. Like terms refer to terms that have the same variable raised to the same power. This means that to be considered like terms, not only must the variables be identical, but their exponents must as well. For instance, in the expression \(8x^3y^2\) and \(2x^3y^2\), both terms are like terms because they both have the variables \(x^3y^2\).
To effectively manipulate polynomials, identify which terms can be combined. In our case, terms with the same variable and exponent combination from each polynomial should be grouped together. This step simplifies the process of addition and subtraction.

Addition of Polynomials

Adding polynomials involves combining like terms to simplify the expression. Given the polynomials \(P1 = 8x^3y^2 - 7x^2y^2 + 7x^2y - 3\) and \(P2 = 2x^3y^2 + x^2y - 1\), you'll add each corresponding set of like terms.
To execute this:

• Add \(8x^3y^2 + 2x^3y^2\) to get \(10x^3y^2\).
• Combine \(-7x^2y^2\) and the terms with \(x^2y^2\) from \(P2\) or zero if none.
• Finally, add or subtract the constant terms \(-3\) and \(-1\).

The result is a simplified polynomial that retains all the important characteristics of the original expressions. It keeps everything in order but just in a simpler form.

Subtraction of Polynomials

Subtracting polynomials requires careful distribution of the negative sign across the polynomial being subtracted. In this exercise, you will subtract \(P3 = 4x^2y^2 + 2x^2y + 8\) from the sum of \(P1\) and \(P2\).
To do so:

• Distribute the negative sign to each term in \(P3\), converting it to \(-4x^2y^2 - 2x^2y - 8\).
• Next, combine these terms with the corresponding like terms from \(P1 + P2\).
• Be attentive to signs, especially since a minus can flip the signs, switching addition to subtraction.

This subtraction helps to reach a more consolidated and accurate expression as we strive to simplify.

Simplification of Expressions

Simplifying expressions is the process of condensing polynomials to their most compact form. After performing addition and subtraction, the final step is to ensure all like terms are efficiently combined.
This means:

• Grouping all terms with similar variables and powers back together.
• Adding or subtracting these to get a single expression.

For instance, in the final result \(10x^3y^2 - 9x^2y^2 + 6x^2y - 12\), you can't simplify the expression further as no terms are like terms anymore. This technique clears up the expression, getting rid of any unnecessary complexity and leaving behind a neat and accessible form.