Problem 35 - Elementary and Intermediate Algebra

Question

Add the polynomials. $$ \left(2 x^{2}+x y+3 y^{2}\right)+\left(5 x^{2}-y^{2}\right) $$

Step-by-Step Solution

Answer

The resulting polynomial is \(7x^2 + xy + 2y^2\).

Step 1: Identify Like Terms

First, identify like terms in the given polynomials. Like terms are those that have the same variables raised to the same power. In this case, the like terms are: - For \(x^2\), the terms are \(2x^2\) and \(5x^2\).- For \(y^2\), the terms are \(3y^2\) and \(-y^2\).- The term \(xy\) appears only in the first polynomial, so it does not have a like term in the second polynomial.

Step 2: Add Coefficients of Like Terms

Now, add the coefficients of like terms:- For \(x^2\), add \(2 + 5 = 7\).- For \(y^2\), add \(3 + (-1) = 2\). This simplifies to \(3 - 1 = 2\).- The term \(xy\) remains unchanged as it has no like term to add.

Step 3: Write the Resulting Polynomial

Combine the results from Step 2 to write the resulting polynomial:- Combine \(7x^2\), \(1xy\), and \(2y^2\).The resulting polynomial is: \[7x^2 + xy + 2y^2\]

Key Concepts

Understanding Like Terms

In the world of polynomials, identifying and understanding like terms is crucial. Like terms are those that contain the same variables raised to the exact same powers. This means they can be combined through addition or subtraction. For example, in the given exercise, the terms \(2x^2\) and \(5x^2\) are like terms because they both contain the variable \(x\) squared. Similarly, \(3y^2\) and \(-y^2\) are like terms as they both include \(y\) squared. However, the term \(xy\) does not have a corresponding like term in the second polynomial; thus, it remains separate. Being able to identify these like terms is the first step to simplifying polynomial expressions effectively.
Create a habit of scanning through the polynomial expression to pair up like terms. It's like finding matching pairs in a game—once you spot them, you're one step closer to solving the puzzle!

The Role of Coefficients in Polynomial Addition

When it's time to combine like terms, you need to focus on the coefficients. Coefficients are the numerical parts of terms that multiply the variables. To simplify polynomials, you'll add these coefficients together for each set of like terms.
For instance, with our like terms \(2x^2\) and \(5x^2\), the coefficients are 2 and 5. Adding these gives \(7x^2\). Similarly, for \(3y^2\) and \(-y^2\), the coefficients 3 and -1 sum up to 2. Thus, it simplifies to \(2y^2\).

• Identify the coefficients in front of each variable power and make sure to operate only on them.
• Add or subtract the coefficients based on their signs.
• Keep the variables and exponents unchanged while combining.

Understanding how to manipulate these coefficients is vital for correctly reducing polynomial expressions into their simplest form.

Forming a Polynomial Expression

Once like terms have been identified and their coefficients combined, the next step is constructing the new expression. A polynomial expression is a sum of multiple terms, each consisting of a coefficient and a variable with a non-negative integer exponent. After adding the coefficients of the like terms, these terms can be written together to form the simplified polynomial.
In our exercise, the combined results of the like terms form the final polynomial \(7x^2 + xy + 2y^2\). The polynomial here is simple and concise:

• \(7x^2\) represents the sum of the two \(x^2\) terms.
• \(xy\) remains unchanged as it has no pair to combine with.
• \(2y^2\) is the result of adding the \(y^2\) terms.

Remember that writing a polynomial is about expressing the entire combination of terms in its simplest form, ensuring clarity and simplicity.
Mastering this allows for an easier understanding and manipulation of algebraic expressions, laying down a solid foundation for more complex algebraic operations.