Problem 11 - College Algebra

Question

In Exercises \(11-16,\) factor by grouping. $$x^{3}-2 x^{2}+5 x-10$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(x^{3}-2 x^{2}+5 x-10\) is \( (x-2)(x^2+5)\).

Step 1: Arrange the polynomial

First, arrange the polynomial terms in decreasing order of powers of x, which is currently done for us in the given polynomial: \(x^{3}-2 x^{2}+5 x-10\).

Step 2: Group the terms

Next, group the first two terms and the last two terms together: \( (x^{3}-2 x^{2}) + (5 x-10)\). We group terms in a way that reduces the degree of the terms and allows us to factorize them.

Step 3: Factorize the groups separately

In each group, identify a common factor that can be factored out. Let's do it for each group separately: In the first group, \(x^2\) is a common factor and factorizing we get: \(x^2 (x - 2)\). In the second group, 5 is a common factor and factorizing we get: \(5 (x - 2)\). Now, our expression looks like this: \(x^2 (x - 2) + 5 (x - 2) \).

Step 4: Factorize the entire expression

Observing the factorized form, we can see in both terms, \((x - 2)\) is a common factor. We can now factorize out \((x - 2)\): which gives \((x-2)(x^2+5)\).

Key Concepts

Polynomial Factorization

Polynomial factorization is a process of breaking down a complex polynomial expression into a product of simpler factors. It is much like finding the 'building blocks' of the expression that, when multiplied together, will give you the original polynomial. This is essential in algebra as it simplifies expressions and can make solving equations easier.

Factoring by grouping, as you saw in the exercise with the polynomial (x^{3}-2 x^{2}+5 x-10), involves rearranging and grouping terms so that each group has a common factor. This is done in a systematic way, starting with arranging the terms in descending powers of x. Once grouped, you can factor out the common factors and simplify the expression further by combining any like terms.

In practice, look for patterns such as pairs of terms that share a common factor. When identified, these common factors become the key to unlocking a more simplified form of the polynomial.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as addition and multiplication). The expression represents a particular number or quantity once the variables are replaced with specific values.

The expression (x^{3}-2 x^{2}+5 x-10) from the exercise is an example of an algebraic expression. In working with algebraic expressions, understanding how to properly manipulate and factor them is crucial, as it allows one to rewrite expressions in equivalent forms.

Throughout the factoring process, you must adhere to the fundamental operations of algebra and the distributive law, which states that a(b + c) = ab + ac. This is used in reverse when we 'factor by grouping' - we take a common factor and essentially distribute it across the grouped terms.

Common Factors

Common factors play a pivotal role in factorization. They are numbers, variables, or algebraic expressions that evenly divide two or more terms without leaving a remainder. Identifying the common factors in algebraic expressions is the first and most critical step in the factorization process.

In the given exercise, x^2 is a common factor of the first group (x^{3}-2 x^{2}), and 5 is a common factor of the second group (5 x-10). Once these are factored out, the remainder of the terms within each group (x - 2) become visibly similar, revealing the hidden structure of the algebraic expression.

Factoring not only simplifies expressions but also enables you to solve equations and understand polynomial functions better. Identifying common factors can turn a seemingly impenetrable expression into a more manageable and solvable equation.