Problem 44 - Elementary Algebra

Question

Factor out the GCF. $$ 8 x 3-12 x 2+2 x $$

Step-by-Step Solution

Answer

The expression factored by the GCF is \( 2x(4x^2 - 6x + 1) \).

Step 1: Expression Representation

Identify the expression to be factored: \[ 8x^3 - 12x^2 + 2x \] This is a polynomial with three terms.

Step 2: Identify the Greatest Common Factor (GCF)

Examine each term to find the GCF. All terms include the factor \( x \). The coefficients are 8, 12, and 2. Find the GCF of these coefficients, which is 2. Therefore, the GCF is \( 2x \).

Step 3: Factor Out the GCF

Rewrite each term by factoring out \( 2x \). 1. From \( 8x^3 \), factor out \( 2x \) to get \( 4x^2 \). 2. From \( -12x^2 \), factor out \( 2x \) to get \( -6x \). 3. From \( 2x \), factor out \( 2x \) to get \( 1 \).

Step 4: Rewrite the Expression

Combine the factored terms to rewrite the expression, pulling the common factor outside the parentheses:\[ 2x(4x^2 - 6x + 1) \] This is the simplification of the original polynomial.

Key Concepts

Polynomials

Polynomials are fundamental elements in algebra, comprising expressions with multiple terms. Each term in a polynomial consists of a coefficient and one or more variables raised to a power. In the given exercise, the expression \( 8x^3 - 12x^2 + 2x \) is a polynomial with three terms. Polynomials can have varying degrees, determined by the highest power of the variable. Here, the degree is 3 due to the term \( 8x^3 \).

• Coefficients: These are the numbers in front of the variable terms. In our case, they are 8, -12, and 2.
• Terms: These are each part of the polynomial expression, such as \( 8x^3, -12x^2, \) and \( 2x \).
• Variables: These represent unknowns and are typically denoted by letters like \( x \).

Grasping the structure of polynomials is crucial for algebraic operations like addition, subtraction, and particularly, factoring.

Factoring

Factoring is a technique used to simplify polynomials by expressing them as a product of simpler expressions. When factoring, especially with polynomials, we aim to find the Greatest Common Factor (GCF) that can be 'taken out' of each term.

• Greatest Common Factor (GCF): It's the largest factor that divides all terms of the polynomial. For \( 8x^3 - 12x^2 + 2x \), the GCF is \( 2x \).
• Factoring Out: To factor out the GCF, divide each term by \( 2x \) to simplify the polynomial. This simplifies the polynomial into a product: \( 2x(4x^2 - 6x + 1) \).

This process not only simplifies the expression but is essential in solving equations and inequalities involving polynomials.

Algebraic Expressions

Algebraic expressions are collections of variables and constants connected by operators, like addition and subtraction. They're the language of algebra, allowing for the representation of real-world problems symbolically. The expression \( 8x^3 - 12x^2 + 2x \) is an example. Understanding their components can simplify complex algebraic problems.

• Simplification: Techniques such as factoring help to reduce expressions to their simplest form, making them easier to work with.
• Operations: Addition, subtraction, multiplication, and division can be performed on algebraic expressions underpinned by the rules of arithmetic.
• Solving Problems: By mastering algebraic expressions, one can solve equations and real-world applications, bridging the gap between theory and practical use.

Algebraic expressions form the backbone of more complex mathematics, including polynomial equations and calculus.