Problem 43 - Elementary Algebra

Question

For the following problems, expand the quantities so that no exponents appear. $$ \left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right) $$

Step-by-Step Solution

Answer

Answer: The expanded form of the given expression is $x^4 - y^4$.

Step 1: Write down the given expressions

Write down the given expression: $$ \left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right) $$

Step 2: Apply the distributive property (FOIL method)

Use the FOIL (First, Outer, Inner, Last) method to multiply the two expressions: $$ (x^{2}\cdot x^{2})+(x^{2}\cdot y^{2})-(y^{2}\cdot x^{2})-(y^{2}\cdot y^{2}) $$

Step 3: Simplify and write the result

Simplify each product and rewrite the result: $$ x^{4}+x^{2}y^{2}-x^{2}y^{2}-y^{4} $$

Step 4: Combine like terms

Combine the like terms to obtain the final result: $$ x^{4}+x^{2}y^{2}-x^{2}y^{2}-y^{4} = x^{4}-y^{4} $$ So, the expression after expansion is: $$ x^{4}-y^{4} $$

Key Concepts

FOIL method

The FOIL method is particularly handy when expanding binomials, as it efficiently guides you through the multiplication process. FOIL stands for First, Outer, Inner, Last, emphasizing the order in which you should multiply each term.

Here's a quick breakdown:

• **First:** Multiply the first terms from each binomial.
• **Outer:** Multiply the outermost terms.
• **Inner:** Multiply the innermost terms.
• **Last:** Multiply the last terms from each binomial.

In the problem given, you apply FOIL to \[(x^{2} - y^{2}) (x^{2} + y^{2})\] as follows:

• **First:** \[x^2 \times x^2 = x^4\]
• **Outer:** \[x^2 \times y^2 = x^2y^2\]
• **Inner:** \[-y^2 \times x^2 = -x^2y^2\]
• **Last:** \[-y^2 \times y^2 = -y^4\]

Each of these products forms part of your expanded expression. Understanding FOIL thoroughly ensures that you can tackle any binomial product with confidence.

Distributive Property

The distributive property is a fundamental concept in algebra that facilitates the multiplication of expressions. It's particularly useful when you have parentheses and need to distribute one term to several others.

The distributive property can be expressed as: \[ a(b + c) = ab + ac \]
When applying it to our particular problem, \[(x^2 - y^2)(x^2 + y^2)\], you're essentially distributing each term in the first binomial to every term in the second.

• **Distribute** \[x^2\] over \[(x^2 + y^2)\], resulting in \[x^2 \cdot x^2 + x^2 \cdot y^2\].
• **Distribute** \[-y^2\] over \[(x^2 + y^2)\], leading to \[-y^2 \cdot x^2 - y^2 \cdot y^2\].

So, through the distributive property, you're able to systematically expand the expression by ensuring no term from one binomial is missed when paired with another. This elimination of parentheses provides clarity and a pathway to the next step: combining like terms.

Combining Like Terms

Combining like terms is the vital last part of simplifying expressions. It involves the addition or subtraction of terms with the same variables and exponents, making complex expressions manageable.

Let's revisit the problem expanded using the distributive property:
\[ x^4 + x^2y^2 - x^2y^2 - y^4 \]
When you apply combining like terms:

• **Locate like terms:** \[x^2y^2\] appears twice, once positive and once negative.
• **Combine them:** As they are opposites, they cancel each other out, resulting in \[0\].

The simplified expression is:
\[x^4 - y^4\].
By focusing on like terms, you save time and effort, as you're cutting out redundancy and honing in on the core components of your expression. Mastering this skill allows for elegant and simplified solutions in algebra.