Problem 61 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(x^{2}+2 x y-y^{2}\right)+\left(5 x^{2}-4 x y+20 y^{2}\right) $$

Step-by-Step Solution

Answer

The result is \( 6x^2 - 2xy + 19y^2 \).

Step 1: Identify the Expressions

We will work with two expressions here: \( \left(x^{2}+2xy-y^{2}\right) \) and \( \left(5x^{2}-4xy+20y^{2}\right) \). We are asked to add these two polynomial expressions.

Step 2: Group Like Terms

To simplify the addition, we should group like terms together. The expressions have like terms involving \( x^2 \), \( xy \), and \( y^2 \).

Step 3: Add Coefficients of Like Terms

We add the coefficients for each type of term as follows:- For \( x^2 \): \( 1x^2 + 5x^2 = 6x^2 \)- For \( xy \): \( 2xy - 4xy = -2xy \)- For \( y^2 \): \( -1y^2 + 20y^2 = 19y^2 \)

Step 4: Write the Resulting Expression

Combine the results from adding the coefficients of like terms. The simplified expression is \( 6x^2 - 2xy + 19y^2 \).

Key Concepts

Like Terms

One of the most essential concepts in polynomial operations is understanding 'Like Terms'. A like term refers to terms in a polynomial that have exactly the same variable part, also known as the same literal factors. These terms can be combined through addition or subtraction because their variable parts match perfectly.
For example, in the polynomial expressions given:

• In \( x^2 + 2xy - y^2 \) and \( 5x^2 - 4xy + 20y^2 \), the terms \( x^2 \) from both expressions are like terms because they have the same variable, \( x^2 \).
• Similarly, the \( xy \) terms are also like terms.
• Finally, the \( y^2 \) terms can be classified as like terms.

Identifying like terms is the first step towards simplifying polynomial expressions, as it allows us to systematically combine them, making further operations more straightforward.

Coefficients

Coefficients are the numerical part of polynomial terms. They play a crucial role in calculations involving polynomials, such as addition and subtraction. Each coefficient is directly linked to its variable or term.
When you look at a polynomial like \( x^2 + 2xy - y^2 \), the coefficients are:

• 1 for \( x^2 \) (often not written explicitly but understood as "1")
• 2 for \( xy \)
• -1 for \( y^2 \)

In polynomial addition, as shown in the exercise, the coefficients of like terms are added or subtracted:

• For \( x^2 \) terms, \( 1 + 5 = 6 \)
• For \( xy \) terms, \( 2 - 4 = -2 \)
• For \( y^2 \) terms, \( -1 + 20 = 19 \)

Understanding coefficients helps simplify expressions correctly by carrying out these operations accurately.

Simplifying Expressions

Simplifying expressions involves reducing a polynomial to its most concise form. This is achieved by combining like terms, as we have in our example. Each combined term is the result of adding or subtracting coefficients of like terms, resulting in a new polynomial that is usually shorter and more manageable.
The process of simplifying begins by recognizing and grouping like terms. For example, consider the polynomials:

• Initial: \( (x^2 + 2xy - y^2) + (5x^2 - 4xy + 20y^2) \)
• Step-by-step combination results in \( 6x^2 - 2xy + 19y^2 \)

Through adding the coefficients of matching terms, we simplify the expression. The final expression \( 6x^2 - 2xy + 19y^2 \) is the simplified form where no further like terms can be combined. Simplifying expressions makes it easier to work with polynomials in both algebraic computations and in real-world problem-solving.