Problem 12 - Elementary and Intermediate Algebra

Question

What is the first step in factoring \(8 y^{2}-16 y z-6 y+12 z ?\)

Step-by-Step Solution

Answer

Factor out the common factor 2 first.

Step 1: Identify Common Factors

Look for any common factors in the expression. In our expression, every term has a common factor of 2. Factor out 2 from the entire expression to simplify it.

Step 2: Rewrite the Expression

After factoring out the common factor of 2, the expression becomes:\[2(4y^2 - 8yz - 3y + 6z)\]

Key Concepts

Common Factors

The idea of common factors is pivotal in the process of simplifying and factoring algebraic expressions. A common factor, in simple terms, is a number or expression that evenly divides two or more terms. Identifying these factors is often the first step when you are tasked with factoring an expression.

Consider our given expression: \(8y^{2} - 16yz - 6y + 12z\). To determine the common factor of the expression, observe each of the terms separately:

• \(8y^{2}\) has factors of 8, \(y \cdot y\)
• \(-16yz\) has factors of -16, y, and z
• \(-6y\) has factors of -6 and y
• \(+12z\) has factors of 12 and z

In this instance, you can see that each term can be divided by 2. Thus, 2 is the common factor. By factoring out the common factor, you are essentially simplifying the expression into a more manageable form.

Factoring Techniques

Once common factors are identified, we can explore factoring techniques to further simplify expressions. The process starts with factoring out the greatest common factor, as seen in our initial step with the expression \(8y^{2} - 16yz - 6y + 12z\), where we factored out the number 2.

The essence of factoring is to break down an expression into a product of simpler expressions. There are various techniques employed, such as:

• Grouping: This technique is pertinent when an expression has four terms. You can group pairs of terms to identify common factors.
• Difference of Squares: Applied when an expression fits the form \(a^{2} - b^{2}\). It factors into \((a + b)(a - b)\).
• Quadratic Trinomials: This involves expressions of the form \(ax^{2} + bx + c\). These are factored into the product of two binomials, \((mx + n)(px + q)\), when possible.

In our example, after factoring out the common factor of 2, the goal is to further simplify the expression within the parentheses through suitable techniques.

Simplifying Expressions

Simplifying expressions is a crucial step in making algebraic problems easier to handle. It allows us to look at complex expressions in a more general and manageable way. Once an expression is simplified, solving equations, understanding graph behaviors, or comparing expressions become more straightforward.

In our given exercise, simplifying is achieved by first identifying and then removing the common factor, leaving us with \(2(4y^2 - 8yz - 3y + 6z)\). This simplified version shows the same relationships between the variables with less complexity. The next steps would involve applying other factoring techniques, like those discussed earlier, to reduce the expression further, if needed.

• The first step in simplifying is always to find common factors.
• Next is to decide whether additional factoring can break down the expression further.

Consistently applying these steps will help maintain simplicity in expressions and equations, facilitating ease in solving and further manipulation.