Problem 48 - Algebra 2

Question

Simplify. $$ \left(a^{3}-b\right)\left(a^{3}+b\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \(a^{6} - b^{2}\).

Step 1: Identify the Expression Type

The expression \((a^{3}-b)(a^{3}+b)\) is a difference of squares of the form \((x-y)(x+y)\), where \(x = a^3\) and \(y = b\).

Step 2: Apply the Difference of Squares Formula

Use the difference of squares formula: \[(x-y)(x+y) = x^2 - y^2\]Substitute \(x = a^3\) and \(y = b\) into the formula: \[(a^3-b)(a^3+b) = (a^3)^2 - b^2\].

Step 3: Simplify the Expression

Compute \((a^3)^2\) to get \(a^{6}\). Therefore, the expression becomes: \[a^6 - b^2\].

Key Concepts

Difference of Squares

The concept of the "difference of squares" is an essential algebraic formula that simplifies expressions of the form \((x-y)(x+y)\).
According to this formula, any expression that looks like \((x-y)(x+y)\) can be simplified to \(x^2 - y^2\).
This works because the middle terms \(-xy\) and \(+xy\) cancel each other out, leaving you with just \(x^2 - y^2\).

For example, when you have \((a^3 - b)(a^3 + b)\), you can see this as \((x - y)(x + y)\) where \(x = a^3\) and \(y = b\).
Applying the difference of squares formula here allows simplifying this complex expression swiftly, resulting in \(x^2 - y^2\), which becomes \((a^3)^2 - b^2\), or simply \(a^6 - b^2\).
This formula is particularly handy in polynomial simplification, allowing you to save time and effort while avoiding unnecessary computations.

Polynomial Simplification

Polynomial simplification is all about reducing complex polynomial expressions into a simpler and more understandable form.
In our example, we started with \((a^3-b)(a^3+b)\), a somewhat complicated expression due to its polynomial nature.
By recognizing it as a difference of squares problem, we can make the simplification process more efficient.

Here's the way to approach polynomial simplification for expressions resembling the difference of squares:

• Identify if the expression fits the pattern \((x-y)(x+y)\).
• Apply the difference of squares formula \(x^2 - y^2\).
• Simplify any resulting terms, such as squaring any variables.

In the example \((a^3-b)(a^3+b)\), identifying \(a^3\) as \(x\) and \(b\) as \(y\), implementation of the difference squares formula helps us directly reach a simplified form \(a^6 - b^2\), making it more manageable and comprehensible.

Algebraic Formulas

Algebraic formulas are the foundational tools in algebra that enable the simplification and solving of various mathematical expressions.
One key formula already discussed is the difference of squares \((x-y)(x+y) = x^2 - y^2\).
This formula is particularly useful for simplifying specific types of polynomials that involve differences and sums of terms.

The effective use of algebraic formulas requires:

• Recognizing patterns within algebraic expressions.
• Knowing where and when to apply these formulas effortlessly.
• Practicing with various expressions to deepen understanding.

The exercise of simplifying \((a^3-b)(a^3+b)\) through the application of the difference of squares formula demonstrates how these formulas can turn a seemingly intricate expression into a straightforward result.
Understanding the applications and limits of each formula enhances one's ability to solve algebra-related challenges quickly and accurately.