Problem 29 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 16 y^{2}-34 y+18 $$

Step-by-Step Solution

Answer

The trinomial factors to \((y - 1)(2)(8y - 9)\).

Step 1: Identify the Trinomial

We are given the trinomial \( 16y^2 - 34y + 18 \). Our goal is to factor it using the grouping method.

Step 2: Multiply and Split the Middle Term

First, find two numbers that multiply to the product of the quadratic term ( 16y^2) and the constant term (18), i.e., 16 * 18 = 288, and add up to the linear term coefficient (-34). These numbers are -16 and -18, because -16 * -18 = 288 and -16 - 18 = -34.

Step 3: Rewriting the Trinomial

Rewrite the middle term using the numbers from the previous step: \[ 16y^2 - 16y - 18y + 18 \].

Step 4: Group the Terms

Group the expression in pairs to factor by grouping: \[ (16y^2 - 16y) + (-18y + 18) \].

Step 5: Factor Out the Greatest Common Factor

Factor the greatest common factor from each group: \[ 16y(y - 1) - 18(y - 1) \].

Step 6: Factor the Common Binomial

Notice the common binomial factor \((y - 1)\) in both groups: \[ (y - 1)(16y - 18) \].

Step 7: Simplify

Further simplify if needed. Recognize that 16y - 18 can be factored: \((16y - 18) = 2(8y - 9)\). So the complete factorization is: \[ (y - 1)(2)(8y - 9) \].

Key Concepts

Grouping Method for Factoring Trinomials

The grouping method is a technique used to factor trinomials, especially when the leading coefficient (the number in front of the highest degree term) is not 1. This involves rewriting the expression in a way that allows you to group terms together and factor by grouping effectively. Imagine this like organizing your clothes so that similar items are together, making it simpler to handle.

In our example, we start with the trinomial:

• Identify the trinomial: Look at the expression you have. In this case, it's \(16y^2 - 34y + 18\).
• Multiply the quadratic term and the constant term: Multiplying the leading coefficient (16) by the constant term (18) gives us 288. This is the key product you need.
• Finding pairs: Think of two numbers that multiply to 288 and add up to the middle term coefficient, which is -34. Here, the numbers -16 and -18 work.
• Rewrite the trinomial: break down the middle term \(-34y\) as \(-16y - 18y\), allowing us to reshape the trinomial to \(16y^2 - 16y - 18y + 18\).
• Group the terms: Organize as \((16y^2 - 16y) + (-18y + 18)\) and factor out from each group. This step is to assemble terms so they reveal a common factor more clearly.

Following the grouping method breaks down the problem into manageable steps, making it simpler to eventually factor the polynomial completely.

Understanding the Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by terms in an expression. Consider it like finding the largest box both your toy car and stuffed animal fit in, allowing them to conveniently stay together. In our factoring trinomial procedure, identifying the GCF is crucial for simplifying grouped terms.

• Find the GCF for the first pair: In \((16y^2 - 16y)\), both terms share the factor \(16y\). You "take out" \(16y\) from the pair, noticing each term is divisible by it, resulting in \(16y(y - 1)\).
• Find the GCF for the second pair: In \((-18y + 18)\), both terms share the factor \(-18\), hence it factors to \(-18(y - 1)\).
• Combined approach: Upon factoring the GCF from each pair, the expression appears as \(16y(y - 1) - 18(y - 1)\), revealing a common binomial factor \((y - 1)\).
• Importance: Recognizing the GCF in individual groups simplifies complex expressions and unveils shared factors, setting the stage for the final factorization.

By identifying and extracting the greatest common factor, you simplify expressions and discover the elements that hold different parts of your polynomial together. It's like shining a light on the common core of parts of your problem.

Quadratic Expressions and Their Components

Quadratic expressions are polynomials where the highest degree is two, forming the backbone of our exercises. Think of these expressions as a box where the size of the box (degree of the polynomial) is crucial to solving it efficiently. The main elements include the quadratic term, the linear term, and the constant. Understanding these pieces helps in approaching problems like factoring with clarity.

• Quadratic term: The term with the squared variable, like \(16y^2\). This term decides the "width" of your polynomial problem. It often is the source of pairing needed calculations when factoring.
• Linear term: Represented here by \(-34y\), it outlines the "direction" of the problem's solution, contributing to the sum needed in methods like grouping.
• Constant term: Simply put, the number without a variable, like 18 in \(16y^2 - 34y + 18\). It roots your equation into certain numerical values.

Each element plays a role in solving quadratic expressions, influencing methodologies like factoring by determining the arithmetic landscape you work within. Identifying and understanding these terms pave the way for handling problems step by step, no different than finding the best tools designed for specific tasks in a toolbox.