Problem 104 - Algebra and Trigonometry

Question

Factor Completely. $$12 x^{2}(x-1)-4 x(x-1)-5(x-1)$$

Step-by-Step Solution

Answer

The completely factored form of the given expression is \((x-1)(12x^2 - 4x - 5)\).

Step 1 - Identify Common Factors

Each term in the given expression \(12 x^{2}(x-1)-4 x(x-1)-5(x-1)\) has \((x-1)\) as a common factor. So, it's clear that factorization can be performed. This step is all about recognizing that.

Step 2 - Factor Out the Common Factor

Remove \((x-1)\) from each term. This is akin to dividing each term by \((x-1)\). This results in the expression: \((x-1)(12x^2 - 4x - 5)\).

Step 3 - Simplify Further If Possible

The expression is now in the form of \((x-1)(ax^2+bx+c)\). You can see if the quadratic expression can be factored further. In this case, it cannot be factored further. So, the completely factored form of the given expression is \((x-1)(12x^2 - 4x - 5)\).

Key Concepts

Common Factors

In the world of algebra, recognizing common factors is an essential skill that simplifies expressions. A common factor is a term that divides two or more terms in an expression without leaving a remainder. Identifying common factors helps in transforming a complex expression into a simplified version, making it easier to understand and solve.

Let's take the expression from the exercise:

• In the expression \(12x^{2}(x-1) - 4x(x-1) - 5(x-1)\), each term shares the factor \((x-1)\).
• The key is to "pull out" or factor out the common term, which drastically reduces the complexity.

This process can be likened to extracting the greatest common divisor in number theory. Pulling out the common factor of \((x-1)\) transforms the entire expression into a cleaner product: \((x-1)(12x^2 - 4x - 5)\). This step makes further working on the equation more manageable.

Quadratic Expression

A quadratic expression is an algebraic expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Such expressions are crucial in algebra because they often appear in various problems, from physics to finance.

In the step-by-step solution, after factoring out the common term, the expression reduces to a quadratic form of \((x-1)(12x^2 - 4x - 5)\):

• Here, \(12x^2 - 4x - 5\) is our quadratic expression.
• The challenge is to see if it can be factored further.

Factoring quadratics typically involves finding two numbers that multiply to the product of \(a\) and \(c\) (here, -60) and add to \(b\) (here, -4). However, since this expression cannot be factored further with integers, it remains as it is.
The ability to recognize when a quadratic is fully factored or needs further solving (perhaps via the quadratic formula) is vital to mastering algebra.

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. These expressions can represent quantities and relationships and are the backbone of algebra.

The original exercise starts with an algebraic expression that includes polynomials and needs simplification. The expression given:

• \(12 x^{2}(x-1) - 4 x(x-1) - 5(x-1)\)

is a combination of terms that can be variously categorized as linear and quadratic, with coefficients and a common factor. Simplifying such expressions typically involves:

• First, identifying any shared elements among terms (as we did with \((x-1)\)).
• Then, reducing the expression step-by-step, prioritizing factors and polynomial identities.

Understanding algebraic expressions is foundational—essential for everything from solving simple equations to evaluating complex algebraic models in higher mathematics. The given exercise offers a fine example of applying these principles to simplify and completely factor polynomials.