Problem 12 - College Algebra Essentials

Question

$$\text { Factor by grouping.}$$ $$x^{3}-3 x^{2}+4 x-12$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(x^{3}-3 x^{2}+4 x-12\) by grouping is \((x-3)(x^{2}+4)\).

Step 1: Group the expression

The first step is to group the terms in pairs, observing the possible common factors. Here, the given polynomial \(x^{3}-3 x^{2}+4 x-12\) can be grouped as \((x^{3}-3 x^{2})+(4x -12)\).

Step 2: Factor by grouping

In each group, factor out the greatest common factor. For the first group \((x^{3}-3 x^{2})\), the common factor is \(x^2\), giving \(x^2(x-3)\). For the second group \((4x-12)\), the common factor is 4, giving \(4(x-3)\).

Step 3: Factoring common binomial

Now observe that the expressions we have factored out from each group, \(x-3\), are the same. This reveals another common factor of \(x-3\) which we then factor out, leaving the factors \(x^2\) and 4. This gives us the final factored form which is \((x-3)(x^{2}+4)\).

Key Concepts

Polynomial Factoring

Polynomial factoring is a crucial skill in algebra that allows you to simplify algebraic expressions and solve complex equations more easily. It involves breaking down a polynomial into the product of its factors, which can be other polynomials or constants. This process is similar to finding what numbers multiply together to give another number, but with algebraic expressions.

For instance, factoring the polynomial \(x^3-3x^2+4x-12\) requires recognizing patterns and applying specific techniques like grouping. By reorganizing the terms into pairs, finding common factors within these groups, and factoring them out, the polynomial can often be expressed as a product of simpler expressions. This makes subsequent computation or solving for variables much more manageable.

Common Factors

Identifying common factors is at the heart of many factoring techniques. A common factor is an algebraic expression that divides exactly into each term of the polynomial. In the grouped polynomial \(x^3-3x^2+4x-12\), we find common factors within the pairs.

In the first group, \(x^3-3x^2\), \(x^2\) is the common factor, while in the second group, \(4x-12\), the number 4 is the common factor. By extracting these common factors, we simplify the polynomial groups into \(x^2(x-3)\) and \(4(x-3)\), respectively. This step reduces the complexity of the polynomial and prepares us for further simplification.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. Consider the factored components such as \(x^2\) and \(x-3\). These are algebraic expressions representing areas of a polynomial that carry certain values or variables.

The expression \(x-3\) appears in both groups after factoring by grouping, illustrating a shared binomial factor that can be factored out further. Algebraic expressions like these are manipulable through arithmetic operations, enabling us to reconstruct or deconstruct polynomials for simplification or problem-solving purposes.

College Algebra

Fundamental Skills

College algebra encompasses a wide array of algebraic concepts, including the factoring of polynomials, simplified through techniques such as grouping. Recognizing patterns and applying algebraic rules to manipulate expressions effectively are fundamental skills cultivated in college algebra.

Application and Problem Solving

This level of mathematical study often focuses on real-world application and problem-solving. For example, factoring \(x^3-3x^2+4x-12\) may come up in analyzing mathematical models, optimizing functions related to physics or economics, or solving for unknown variables. The ability to factor by grouping, therefore, represents more than just a technique—it is a gateway to more advanced mathematical comprehension and application.