Problem 54 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(x^4 + 2x^2y + y^2 - 4\).

Step 1: Recognize the Given Expression

We need to simplify the expression \((x^{2}+y-2)(x^{2}+y+2)\). Notice that this is a multiplication of two binomials.

Step 2: Apply the Difference of Squares Formula

Recognize this multiplication as a difference of squares. It takes the form \((a-b)(a+b)\) where \(a = x^2 + y\) and \(b = 2\). The formula is \((a-b)(a+b) = a^2 - b^2\).

Step 3: Substitute Values into the Formula

Substitute \(a = x^2 + y\) and \(b = 2\) into the formula. This gives us \((x^2 + y)^2 - 2^2\).

Step 4: Simplify Each Term

First, compute the square of \(a\): \((x^2 + y)^2 = (x^2)^2 + 2(x^2)(y) + y^2 = x^4 + 2x^2y + y^2\). Then, compute the square of \(b\): \(2^2 = 4\).

Step 5: Write the Simplified Expression

Combine the results to get \(x^4 + 2x^2y + y^2 - 4\). This is the simplified form of the original expression.

Key Concepts

Binomials

In algebra, a binomial is a polynomial with exactly two terms. These terms are usually connected by either a plus or a minus sign. For example, in the expression \((x^2 + y - 2)\), the terms \(x^2 + y\) and \(-2\) are connected by subtraction, making it a binomial.

Binomials are a common occurrence in algebraic expressions and are the building blocks for many operations, such as polynomial multiplication. Understanding binomials is crucial because they set the stage for more complex manipulations and operations.

When dealing with binomials, especially in multiplication, it's useful to recognize certain patterns, such as the difference of squares. Binomials are versatile and appear in various mathematical problems, helping simplify and solve expressions effectively.

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials together. A polynomial can have multiple terms, but when multiplying simpler versions like binomials, we use specific techniques such as the distributive property.

For example, consider the multiplication of two binomials: \((x^2 + y - 2)(x^2 + y + 2)\). Instead of directly multiplying each term, recognizing patterns like the difference of squares allows us to simplify the process. Here, this expression fits the pattern because we can rearrange it as \((a-b)(a+b)\).

• Identifying the terms involved: \(a = x^2 + y\) and \(b = 2\).
• Using the formula: \((a-b)(a+b) = a^2 - b^2\).
• This prevents errors and makes calculations more straightforward.

By understanding polynomial multiplication and recognizing these patterns, simplifying complex expressions becomes more manageable.

Simplification in Algebra

Simplification in algebra refers to reducing expressions to a more concise form without changing their value. It's an essential skill that ensures problems are easier to work with and understand.

In the context of the problem \((x^2 + y - 2)(x^2 + y + 2)\), simplification is achieved by using the difference of squares method. Here's how you apply it:

• Apply the difference of squares pattern, \(a^2 - b^2\).
• Calculate each square separately before combining them:
  • \((x^2 + y)^2 = x^4 + 2x^2 y + y^2\)
  • \(2^2 = 4\)
• Combine these results to obtain the simplified expression: \(x^4 + 2x^2 y + y^2 - 4\).

Understanding how to simplify expressions through such formulas not only makes problems easier but also sharpens algebraic skills, providing techniques that are useful in more advanced topics of mathematics.