Problem 18 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ 8 y z-6 z-12 y+9 $$

Step-by-Step Solution

Answer

The factored form is \((2z - 3)(4y - 3)\).

Step 1: Group the Terms

Start by grouping the terms in pairs that can potentially be factored by a common factor. The expression is: \[ 8yz - 6z - 12y + 9 \]Group it as follows: \[ (8yz - 6z) - (12y - 9) \]

Step 2: Factor Out the Greatest Common Factor from Each Group

Identify and factor out the greatest common factor from each group. For the first group \((8yz - 6z)\), factor out \(2z\): \[ 2z(4y - 3) \]For the second group \((12y - 9)\), factor out \(-3\): \[ -3(4y - 3) \]

Step 3: Factor Out the Common Binomial Factor

Notice that \((4y - 3)\) is a common factor in both groups:The expression can now be written as:\[ (2z - 3)(4y - 3) \]

Step 4: Check the Factored Form

Verify that the factored form multiplies back to the original expression \(8yz - 6z - 12y + 9\) to ensure accuracy. Expanding \((2z - 3)(4y - 3)\):- \(2z \times 4y = 8yz\)- \(2z \times -3 = -6z\)- \(-3 \times 4y = -12y\)- \(-3 \times -3 = 9\)Combine: \(8yz - 6z - 12y + 9\). This matches the original expression.

Key Concepts

Understanding the Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is a key concept when factoring polynomials. It involves identifying the largest factor that is common to each term in the polynomial. For example, in the polynomial \(8yz - 6z - 12y + 9\), each term has its own set of factors. By isolating parts of the expression such as \(8yz - 6z\) and \(12y - 9\), the first step is to look at these "groups" separately.

• For the group \(8yz - 6z\), the GCF is \(2z\), since both terms share these factors.
• For \(12y - 9\), the GCF is \(-3\) because these are common factors they share.

Factoring out the GCF of each group simplifies the expression and makes it easier to identify further common factors. This stage is crucial because it reduces the polynomial to its simplest components, making other factorization methods more straightforward.

Exploring Binomial Factors

When working with polynomials, identifying binomial factors is an essential skill. A binomial factor is essentially a two-term algebraic expression, which can be repeated in a polynomial structure. In the initial expression \(8yz - 6z - 12y + 9\), after factoring out the greatest common factor from each group, you notice a repeated binomial factor.

In this case, once you have factored \(2z(4y - 3)\) and \(-3(4y - 3)\) from each grouping, you observe \((4y - 3)\) is common in both parts. Having a common binomial factor allows these terms to be expressed in a simpler product form:

• \((2z - 3)(4y - 3)\)

Recognizing and utilizing these binomial factors is advantageous because it breaks down complex polynomials into simpler, more manageable expressions. This makes multiplication and further factorization simpler and efficient.

The Grouping Method as a Factoring Technique

The grouping method is a strategic approach in factoring polynomials, especially useful when dealing with four-term expressions like \(8yz - 6z - 12y + 9\). This method involves rearranging and grouping terms to facilitate easier factorization.

Steps to apply the grouping method include:

• First, split the polynomial into two pairs: \((8yz - 6z)\) and \(- (12y - 9)\).
• Factor each pair separately to find the greatest common factor, resulting in \(2z(4y - 3)\) and \(-3(4y - 3)\).
• Identify any common factors between these groups, like \((4y - 3)\).
• Finally, use this common factor to combine the groups into a simpler form, \((2z - 3)(4y - 3)\).

The grouping method is straightforward yet powerful, providing a reliable means to factor complex expressions by systematically breaking them down. This step-by-step approach ensures each component is addressed, leading to a neatly factored polynomial that matches the original when expanded.