Problem 82 - Elementary Algebra

Question

Factor by grouping. $$ x 3-3 x 2+5 x-15 $$

Step-by-Step Solution

Answer

The factored form is \((x - 3)(x^2 + 5)\).

Step 1: Identify groups

Given the polynomial equation \(x^3 - 3x^2 + 5x - 15\), notice that we can try to group the terms into two pairs: \((x^3 - 3x^2)\) and \((5x - 15)\). This approach helps to factorize by grouping.

Step 2: Factor out the greatest common factor in each group

Within the first group, \(x^3 - 3x^2\), the greatest common factor is \(x^2\). Factoring it out gives us \(x^2(x - 3)\). For the second group, \(5x - 15\), the greatest common factor is \(5\). Factoring it out gives us \(5(x - 3)\).

Step 3: Combine as a common factor

Both expressions \(x^2(x - 3)\) and \(5(x - 3)\) have the common factor \((x - 3)\). Factor this common term out to get \((x - 3)(x^2 + 5)\).

Step 4: Verify the factorization

Expand the factored expression \((x - 3)(x^2 + 5)\) to ensure it matches the original polynomial. Distribute \((x - 3)\) across \((x^2 + 5)\) to get \(x^3 - 3x^2 + 5x - 15\), which confirms the factorization is correct.

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. It's similar to factoring numbers, where you break down a number into its prime components. In the context of algebra, the factors are algebraic expressions instead of numbers. The aim is to simplify polynomials for easier computation or to solve polynomial equations.

Factorization by grouping is one common technique used when polynomials have four or more terms. This method looks for ways to group terms and factor them separately, which can often reveal a common binomial factor. By recognizing and applying this technique to the polynomial \(x^3 - 3x^2 + 5x - 15\), we discovered the factorization \((x - 3)(x^2 + 5)\).

The goal is always to make complex expressions simpler, uncover hidden symmetries within polynomials, or allow for a clearer path to solving the equation. Polynomial factorization is a powerful tool in algebra that provides clarity and utility in many mathematical problems.

Common Factor

A common factor is a number or expression that divides each term in an equation without leaving a remainder. Finding common factors is often a critical step when factoring polynomials, like in the original exercise \(x^3 - 3x^2 + 5x - 15\).

To successfully factor by grouping, we must identify the greatest common factor (GCF) within each group of terms:

• For \(x^3 - 3x^2\), the GCF is \(x^2\), which results in \(x^2(x - 3)\).
• For \(5x - 15\), the GCF is \(5\), which leads to \(5(x - 3)\).

Both expressions in the example polynomial end up sharing a common factor of \((x - 3)\). By factoring this term out, the polynomial simplifies greatly. Recognizing and extracting common factors are essential skills in simplifying algebraic expressions and are fundamental to solving polynomial equations efficiently.

Algebraic Expressions

Algebraic expressions are combinations of terms formed by variables and coefficients, which are constants or numbers. Each expression can have variables raised to powers and can include operations such as addition, subtraction, multiplication, and division.

In the polynomial \(x^3 - 3x^2 + 5x - 15\), there are four terms, and each can be thought of as a separate algebraic expression. To factor efficiently, it's crucial to understand how these terms interact and can be rearranged or combined.

The strategy of grouping provides a way to identify sub-expressions that can be simplified using common factors. This polynomial illustrates that even complex-looking expressions can have simpler forms found through thoughtful analysis. Working with algebraic expressions requires practice in operations and in discerning relationships between terms, paving the way for deeper insights into mathematics.