Problem 6 - College Algebra Essentials

Question

$$\text { Factor out the greatest common factor.}$$ $$6 x^{4}-18 x^{3}+12 x^{2}$$

Step-by-Step Solution

Answer

The factored expression of the polynomial \(6x^4 - 18x^3 + 12x^2\) with its greatest common factor factored out is \(6x^2(x^2 - 3x + 2)\).

Step 1: Identify the Greatest Common Factor

First, the greatest common factor in the polynomial \(6x^4 - 18x^3 + 12x^2\) needs to be identified. The numerical coefficients of these terms are 6, -18, and 12. The greatest common factor of these coefficients is 6. Additionally, each term of the polynomial has an 'x' component - \(x^4, x^3\), and \(x^2\). The largest number of 'x's that can be factored out from each term is \(x^2\). Hence, the greatest common factor is \(6x^2\).

Step 2: Divide Each Term by the Greatest Common Factor

In this step, each term of the polynomial is divided by the greatest common factor \(6x^2\). So \(6x^4 / 6x^2\) equals \(x^2\), \(-18x^3 / 6x^2\) equals \(-3x\), and \(12x^2 / 6x^2\) equals 2.

Step 3: Write the Factored Expression

Finally, the factored polynomial expression can be written as \(6x^2\) times the result of Step 2. This creates the expression \(6x^2(x^2 - 3x + 2)\), which is the solution.

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is essential when factoring polynomials. It is, put simply, the highest number that divides exactly into two or more numbers. When dealing with polynomials, we take the concept a step further by looking for the greatest exponent of variables as well as numerical coefficients that are common to each term. Take for example the expression \(6x^4 - 18x^3 + 12x^2\). The GCF here is \(6x^2\), as 6 is the highest common numerical factor, and the squared term is the highest power of x that is a factor of each term in the polynomial. Recognizing the GCF allows us to factor the polynomial into a simpler form.

Polynomial Division

When you've identified the GCF of a polynomial, the next step is to divide each term by the GCF. This process is known as polynomial division and helps to simplify the algebraic expression considerably. In our given exercise, after identifying \(6x^2\) as the GCF, each term of \(6x^4 - 18x^3 + 12x^2\) is divided by \(6x^2\) to yield \(x^2\), \( -3x\), and 2. It’s a process akin to reducing fractions to their simplest form; divide to decrease complexity while retaining the expression’s true value.

Algebraic Expressions

Algebraic expressions compose math entities that can consist of constants, variables, and algebraic operations (addition, subtraction, multiplication, and division). For example, \(x^2 - 3x + 2\) is an algebraic expression resulting from our polynomial division step. What is crucial in understanding algebraic expressions is recognizing not just their components but also the relation between these components, which dictates the structure of the expression and, consequently, its simplification or factorization.

Polynomial Factoring Steps

To master polynomial factoring, one should follow structured factoring steps for clarity and to avoid errors. Initially, seek out the GCF of the terms in the polynomial. Once identified, perform polynomial division to simplify the terms. After these processes, rewrite the original polynomial as a product of the GCF and the result of the division step, as we did with \(6x^2(x^2 - 3x + 2)\). By taking these steps methodically, one can transform complex polynomials into products of simpler factors, facilitating further algebraic manipulation and a deeper understanding of the structure of the expression.