Problem 35 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 4 x-8 y+4 $$

Step-by-Step Solution

Answer

The factored expression is \(4(x - 2y + 1)\).

Step 1: Identify the Terms

Write down each term from the polynomial. The polynomial is \(4x - 8y + 4\). Thus, the terms are \(4x\), \(-8y\), and \(4\).

Step 2: Find Common Factor

Determine the greatest common factor (GCF) of the coefficients 4, -8, and 4. The GCF of 4, 8, and 4 is 4.

Step 3: Factor Out the GCF

Divide each term by the GCF. The term \(4x\) becomes \(x\), \(-8y\) becomes \(-2y\), and \(4\) becomes \(1\) after dividing by 4.

Step 4: Write the Factored Expression

Combine the terms into the expression with the GCF factored out. The factored form is \(4(x - 2y + 1)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is an essential mathematical concept, especially when working with polynomials. To grasp this concept, think of the GCF as the largest number that divides each term of the expression without leaving a remainder. It serves as a building block for simplifying polynomial expressions.
When identifying the GCF, focus on the coefficients, which are the numerical parts of terms. Let's consider our example polynomial: \(4x - 8y + 4\). The coefficients are 4, -8, and 4. To find their GCF, list out the factors of each:

• Factors of 4: 1, 2, 4
• Factors of 8 (consider absolute value): 1, 2, 4, 8

The largest number common in all sets of factors is 4, thus making it the GCF. Understanding this concept helps in streamlining complex expressions into more manageable forms.
Always begin by identifying the GCF when factoring polynomials, as it sets the stage for simplifying the entire expression.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of terms with variables raised to whole number powers. These terms are separated by addition or subtraction operations. In the polynomial \(4x - 8y + 4\), each part such as \(4x\), \(-8y\), and \(4\) is a term.
The exponents in polynomial expressions are usually whole numbers. In our example, \(4x\) features an exponent of 1 on \(x\), even if it's not shown explicitly. Polynomials can have one or more variables and the terms can include constants as seen in the term 4. The structure of polynomials allows them to model diverse mathematical phenomena.
When dealing with polynomials, it's crucial to understand the properties of each term:

• **Coefficient**: The numerical factor of a term. For \(4x\), it's 4.
• **Variable**: Usually represented by letters such as \(x\) or \(y\).
• **Constant term**: A term without a variable, like the 4 in the polynomial.

Recognizing these elements helps break down complex algebraic expressions into simpler components for easier manipulation.

Factored Form

Factoring transforms a polynomial expression into a product of simpler expressions. The factored form displays a clearer understanding of the polynomial’s structure and provides a streamlined expression. This form is particularly useful in solving equations and simplifying expressions.
In the given example, we're tasked with factoring \(4x - 8y + 4\). After determining the GCF to be 4, we "factor out" this GCF from each term:

• For \(4x\), dividing by 4 gives \(x\).
• For \(-8y\), dividing by 4 results in \(-2y\).
• For the constant 4, dividing by 4 produces 1.

Therefore, the factored form of the polynomial is \(4(x - 2y + 1)\). This process simplifies the polynomial, highlighting its underlying components by grouping them through a common factor.
The factored form helps in solving equations because it provides insights into the roots, or solutions, of polynomial equations efficiently. Understanding this form is a step towards mastering algebraic manipulation and problem solving.