Problem 103 - College Algebra

Question

Perform the indicated operations. $$(x+y)(x-y)\left(x^{2}+y^{2}\right)

Step-by-Step Solution

Answer

The simplified product of \((x+y)\), \((x-y)\), and \(\left(x^{2}+y^{2}\right)\) is \(x^{4} - y^{4}\).

Step 1: Apply the Difference of Squares Formula

First, we multiply the two binomials \((x+y)\) and \((x-y)\) using the difference of squares formula. Thus, the result is \((x^{2} - y^{2})\).

Step 2: Multiply the Result with Remaining Binomial

Then, we multiply the result from the previous step with the remaining binomial \(\left(x^{2}+y^{2}\right)\). This gives us the product \((x^{2} - y^{2})(x^{2} + y^{2})\).

Step 3: Apply the Difference of Squares Formula Again

Lastly, we again apply the difference of squares formula to simplify \((x^{2} - y^{2})(x^{2} + y^{2})\) into \(x^{4} - y^{4}\).

Key Concepts

Difference of Squares

The difference of squares is a specific pattern in algebra where we exploit the fact that the square of a number minus the square of another number can be written as \[(a^2 - b^2) = (a - b)(a + b)\].
This is particularly useful in simplifying expressions and making algebraic calculations more straightforward. In our exercise, the expression \((x+y)(x-y)\) follows this pattern. This step illustrates the application of the difference of squares formula, converting it into \(x^2 - y^2\). This formula is a powerful tool in algebraic manipulations:

• It simplifies expressions into a more manageable form.
• Aids in factoring complex algebraic expressions.
• Makes further calculations simpler and more efficient.

To master this pattern, practice recognizing it in different expressions and situations.
Once you're comfortable, you’ll be able to apply it quickly and with confidence.

Binomial Multiplication

Binomial multiplication involves expanding the product of two binomial expressions. When multiplying binomials, each term from the first binomial is multiplied by every term of the second binomial.
This process is often referred to as distributing or using the FOIL method (First, Outside, Inside, Last). But in our given exercise, since we are applying the difference of squares, binomial multiplication simplifies into another straightforward expression, eliminating unnecessary steps.
After multiplying \((x + y)(x - y)\) to get \(x^2 - y^2\), we then multiply this result by a new binomial \((x^2 + y^2)\).
Here, you don't need FOIL for every step.
Thanks to recognizing patterns like difference of squares, the simplification becomes direct and easier.

• It ensures that you arrive at the correct factored or expanded form.
• Makes algebra less about calculation and more about recognizing patterns.

Through practice, anyone can master these multiplication techniques and apply them efficiently.

Algebraic Expressions

Algebraic expressions form the backbone of algebra and encompass a combination of numbers, variables, and operations. They allow us to generalize and solve real-world problems mathematically. Expressions like those in our exercise -- \((x+y)(x-y)(x^2+y^2)\) -- demonstrate how dynamic and interlinked different parts of algebra can be.
Breaking complex expressions into familiar operations, like simplification and multiplication, is key in handling them effectively.Understanding algebraic expressions involves:

• Identifying individual components (like terms and coefficients).
• Recognizing how operations affect the whole expression.
• Applying known formulas, such as the difference of squares, to simplify or rearrange expressions.

Mastering algebraic expressions helps students develop critical problem-solving skills and logical thinking.
Engaging in exercises involving these elements strengthens one’s ability to navigate through increasingly complex mathematical challenges. By grasping these core algebraic concepts, students set a solid foundation for future mathematical success.