Problem 1 - Elementary Algebra

Question

Determine the missing factor. $$ 12 x 3-24 x 2+4 x=4 x(\quad ? \quad) $$

Step-by-Step Solution

Answer

The missing factor is \(3x^2 - 6x + 1\).

Step 1: Identify Common Factors

We start by identifying the terms on the left side of the equation: \(12x^3 - 24x^2 + 4x\). Notice that each term contains an \(x\), which means \(x\) is a common factor. Additionally, the numbers 12, 24, and 4 share a common factor, which is 4.

Step 2: Factor Out the Greatest Common Factor

Factor out the greatest common factor (GCF) from each term. The GCF of the terms \(12x^3\), \(-24x^2\), and \(4x\) is \(4x\). When we factor \(4x\) out, we need to divide each term by \(4x\): \[12x^3 \div 4x = 3x^2, \quad -24x^2 \div 4x = -6x, \quad 4x \div 4x = 1.\]

Step 3: Write the Factored Expression

After factoring out \(4x\), we write the original expression \(12x^3 - 24x^2 + 4x\) as \(4x(3x^2 - 6x + 1)\).

Step 4: Compare with Given Expression

We compare the expression \(4x(3x^2 - 6x + 1)\) with the provided expression \(4x(\quad ? \quad)\). It is evident that the missing factor is the polynomial inside the parentheses: \(3x^2 - 6x + 1\).

Key Concepts

Greatest Common Factor

In the realm of algebra, the "Greatest Common Factor" (GCF) refers to the largest number or term that can evenly divide all the terms in an algebraic expression. Finding the GCF is a fundamental process when simplifying expressions, especially in factoring polynomials.

To determine the GCF in a polynomial, look for both numerical and variable components shared by each term. For example, in the expression \(12x^3 - 24x^2 + 4x\), we can see that:

• The coefficients 12, 24, and 4 have a GCF of 4.
• The variable component is shared as an \(x\) in each term.

This means \(4x\) is the GCF because it is the largest term that divides each component of the expression. Factoring out the GCF simplifies the entire expression, revealing underlying polynomial structures.

Understanding how to identify and factor out the GCF is crucial in algebra for simplifying expressions and solving equations efficiently.

Algebraic Expressions

Algebraic expressions are a way to represent mathematical ideas using numbers, variables, and operations. They form the foundation of algebra, allowing you to model real-world situations mathematically and solve problems.

In expressions like \(12x^3 - 24x^2 + 4x\), each part is known as a term. Terms consist of:

• A coefficient (a number in front of the variable).
• One or more variables raised to an exponent.

Algebraic expressions can be manipulated and simplified through various operations, including addition, subtraction, multiplication, division, and factoring. Learning these techniques is essential for making sense of complex equations and finding solutions.

When dealing with polynomials, which are a type of algebraic expression defined by multiple terms, understanding how to factor effectively is key. This includes identifying patterns and common factors, which make it easier to simplify or solve expressions further.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial expression to its simplest form. This is done by factoring out the greatest common factor and combining like terms to make the expression more concise and manageable.

Take the polynomial \(12x^3 - 24x^2 + 4x\). The simplification process starts with identifying the greatest common factor, \(4x\), as we demonstrated in previous steps.

Once you factor out \(4x\), the expression becomes \(4x(3x^2 - 6x + 1)\). Now, it's in a more simplified form, easier to work with or solve in equations:

• You’ve reduced the original polynomial to fewer, simpler terms.
• The resulting expression is perfect for further analysis, plugging into equations, or integration into more complex algebraic methods.

By mastering polynomial simplification, you make solving algebra problems quicker and more efficient, clearing the path for more advanced mathematics.