Problem 84 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The factored form of the given expression by grouping is \((x^2 - 3)(4x - 1)\).

Step 1: Grouping

Group the given expression into two separate expressions by arranging the given expression in descending order and grouping associated terms together. We can rewrite it in the form of \(4x^3 - x^2\) and \(-12x + 3\) .

Step 2: Factor out Common Terms

Factor out the common polynomial as much as possible from the grouped expressions. The expression \(4x^3 - x^2\) can be factored by \(x^2\), and \(-12x + 3\) can be factored by \(-3\), obtaining the terms \(x^2(4x - 1)\) and \(-3(4x - 1)\).

Step 3: Factoring the Common Binomials

Factor out the common binomial term (\(4x - 1)\) of both expressions, obtaining \((x^2 - 3)(4x - 1)\).

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that involve variables raised to whole number powers. They can include constants and coefficients but do not contain division by a variable, square roots, or variables in the denominator. Polynomials are fundamental in algebra because they appear in equations and inequalities frequently.

Examples of polynomial expressions include:

• Single-variable: Such as \[3x^2 + 2x + 1\]
• Multi-variable: Such as \[4x^3 - y + z^2\]

Understanding these expressions is crucial, as they are the building blocks for many algebraic methods like factoring and solving equations.

Factoring Techniques

Factoring is a technique used to simplify expressions. It involves breaking down complex polynomials into simpler pieces, called factors, which can multiply together to form the original expression. This is incredibly useful for solving equations and simplifying fractions.

There are several common factoring techniques:

• Grouping: Used when an expression can be separated into distinct parts that can be individually factored.
• Greatest Common Factor (GCF): Factoring out the largest factor common to all terms.
• Difference of Squares: Used when expressions fit the form \[a^2 - b^2\], which factors into \[(a+b)(a-b)\].

Understanding the basics of factoring makes solving algebraic expressions far easier, especially when combined in a method like factoring by grouping.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators such as addition, subtraction, multiplication, and division. These expressions lay the groundwork for algebra, allowing us to model real-world situations mathematically. Each part of an algebraic expression, like coefficients and terms, holds vital information about the expression's overall meaning.

Algebraic expressions can be simple or complex:

• Simple: Consists of a single term like \[7x\].
• Complex: Consists of multiple terms, such as \[4x^{3} - x^{2} - 12x + 3\], each term separated by '+' or '-' signs.

Recognizing algebraic expressions and their parts enables students to apply operations such as factoring, expanding, and simplifying.

Common Factors

In mathematics, a common factor is a number or term that divides exactly into two or more numbers or expressions. Identifying common factors in polynomial expressions is essential for simplifying and factoring them.

Steps to find common factors:

• List the factors for each term in the expression.
• Identify the largest factor that appears in every list.
• Use this common factor to factor out from the expression.

In the example given \[4x^3 - x^2 - 12x + 3\], common factors were used in grouping terms in pairs to simplify the problem by taking out \[x^2\] from one group and \[-3\] from the other, revealing the shared binomial \[(4x - 1)\]. This process of determining and using common factors is key in refining algebraic expressions.