Problem 67 - Introductory Algebra for College Students

Question

Factor completely. $$12 x^{2}-12 x+3$$

Step-by-Step Solution

Answer

The factorized form of the equation \( 12x^{2} - 12x + 3 \) is \( (3x - 3)(4x - 1) \).

Step 1: Identify the coefficients

Identify the coefficients in the equation. Here, the coefficients are 12, -12, and 3 representing the \( a \), \( b \), \( c \) respectively.

Step 2: Find the factors that satisfy the condition

Find two numbers that multiply to 12 (the coefficient of \( x^{2} \)) and add up to -12 (the coefficient of \( x \)). These numbers are -3 and -4.

Step 3: Express the equation in terms of these factors

Rewrite the middle term using these numbers and factor by grouping. The equation becomes: \( 12x^{2} - 3x - 9x + 3 \). This can be factored as: \( 3x(4x - 1) -3(4x - 1) \)

Step 4: Simplify the expression to get the factorized form

The final simplification leads to the factored form: \( (3x - 3)(4x - 1) \). Resulting in this, completes the process of fully factoring the equation.

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two. They typically have the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The general feature of quadratic expressions is the \( x^2 \) term, which signifies it's a second-degree polynomial.
Understanding how to manipulate and factor these expressions is crucial for solving related equations and graphing quadratic functions. Quadratic equations are at the core of many problems because they describe parabolic curves. These curves appear frequently in various fields such as physics and engineering.
In the given exercise, the quadratic expression is \(12x^2 - 12x + 3\), with specific coefficients extracted from its standard form.

Polynomial Factorization

Polynomial factorization is a process that involves breaking down a complex polynomial into a product of simpler polynomials or constants. This can make them much easier to solve or manipulate. It is like reversing the multiplication process that originally created the polynomial.
To factor the given quadratic expression \(12x^2 - 12x + 3\), we start by expressing the middle term \(-12x\) in a way that reveals its factors. The factors are numbers that multiply to give the same product as the coefficient of the \(x^2\) term.

• Step 1: Look for numbers that multiply to \(12\) (from \(12x^2\)) and add to \(-12\) (the coefficient of \(x\)). These are \(-3\) and \(-4\).
• Step 2: Rewrite the original expression as \(12x^2 - 3x - 9x + 3\), which allows grouping for factorization.
• Step 3: Factor by grouping, resulting in \(3x(4x - 1) - 3(4x - 1)\).

Finally, simplify it to \((3x - 3)(4x - 1)\). Proper factorization like this helps in understanding the roots of the polynomial and simplifies solving equations.

Algebraic Manipulation

Algebraic manipulation refers to rearranging and simplifying expressions using a set of algebraic rules. It's a vital skill to manage expressions, solve equations, and work through problems effectively.
In the process of factoring our quadratic expression, algebraic manipulation involves re-expressing terms to reveal common factors and simplifying the equation step by step. Breaking down the original expression \(12x^2 - 12x + 3\) into manageable components helps significantly:

• Rewrite terms to prepare for factorization. Here, \(-12x\) is written as \(-3x - 9x\).
• Use grouping to identify common binomials, such as \(4x - 1\).
• Finally, factor out common terms to arrive at \((3x - 3)(4x - 1)\).

Each of these steps demonstrates the power and necessity of algebraic manipulation. It is used to simplify and solve not only quadratic expressions but a wide range of algebraic problems.