Problem 91 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ a^{3}-a^{2} b+a b^{2}-b^{3} $$

Step-by-Step Solution

Answer

The polynomial factors as \((a - b)(a^2 + b^2)\).

Step 1: Recognize Groupable Terms

The given polynomial is: \(a^3 - a^2b + ab^2 - b^3\). Notice that it can be grouped into pairs: \( (a^3 - a^2b) \) and \( (ab^2 - b^3) \). Our plan is to factor each of these groups separately, which will simplify the expression.

Step 2: Factor Each Group

Within the first group \(a^3 - a^2b\), we can factor out \(a^2\) as a common factor, resulting in \(a^2(a - b)\). In the second group \(ab^2 - b^3\), \(b^2\) is the common factor, and factoring it out yields \(b^2(a - b)\). This gives us the expression: \(a^2(a - b) + b^2(a - b)\).

Step 3: Factor by Grouping

Notice that \((a - b)\) is a common factor in both terms of the expression \(a^2(a - b) + b^2(a - b)\). We factor \((a - b)\) out, resulting in \((a - b)(a^2 + b^2)\).

Step 4: Confirm the Full Factorization

The polynomial \(a^3 - a^2b + ab^2 - b^3\) is now rewritten as \((a - b)(a^2 + b^2)\). This is the complete factorization, as neither \((a - b)\) nor \((a^2 + b^2)\) can be factored further over the real numbers.

Key Concepts

Grouping Method

The grouping method is a technique used to simplify polynomial expressions by organizing similar terms. It’s quite like organizing socks, where you pair similar ones together.
In the polynomial \(a^3 - a^2b + ab^2 - b^3\), the terms are arranged such that grouping makes it easier to factor them. We begin by creating groups of terms that share a factor.
- Group the terms: \((a^3 - a^2b)\) and \((ab^2 - b^3)\).
- Each group can be simplified further, which is a critical step for further factorization.

This method is a handy tool for tackling polynomial factorization challenges, especially when factors aren't immediately obvious.

Common Factor

Identifying and extracting a common factor from groups of terms is like finding a thread that ties socks together. It is an essential step to simplify any polynomial.
For our polynomial \(a^3 - a^2b + ab^2 - b^3\), identifying common factors was key in the solution process.
- In the group \((a^3 - a^2b)\), the common factor is \(a^2\), which simplifies this group to \(a^2(a-b)\).
- For \((ab^2 - b^3)\), the common factor is \(b^2\), yielding \(b^2(a-b)\).

By factoring out these common pieces, the expression becomes much simpler to manage. This reduction is a crucial part of algebraic manipulation.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operations that create mathematical statements. In the polynomial \(a^3 - a^2b + ab^2 - b^3\), each term combines the variables \(a\) and \(b\) with different powers and coefficients.
Understanding what an algebraic expression is helps determine how you manipulate it.
- Terms are ordered in specific sequences, sometimes making patterns observable.
- Operations between terms are crucial, as they guide you in applying strategies like the regrouping and factorization.

Comprehending expressions as mathematical sentences is vital for decoding the factoring or simplification methods needed for solving.

Complete Factorization

Achieving complete factorization means breaking down a polynomial into product terms that cannot be further simplified over the real numbers. For \(a^3 - a^2b + ab^2 - b^3\), the complete factorization achieved is \((a-b)(a^2 + b^2)\).
Ensure all common factors are identified and extracted across the entire expression.
- When factoring out \((a-b)\), we simplified the polynomial into two distinct factors: \((a-b)\) and \((a^2 + b^2)\).
- Check if any of the resulting factors can be broken down further, in this case, neither \((a-b)\) nor \((a^2+b^2)\) can be.

Reaching complete factorization is like solving a jigsaw puzzle where you’ve pieced together all components into their precise places.