Problem 10 - A Graphical Approach to College Algebra

Question

Factor the greatest common factor from each polynomial. $$(3 z+2)(z+4)-(z+6)(z+4)$$

Step-by-Step Solution

Answer

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

Step 1: Identify Common Factors

First, look for common factors in the expression \((3z+2)(z+4) - (z+6)(z+4)\). Notice that the term \((z+4)\) is common in both expressions.

Step 2: Factor Out the Common Factor

Factor out the common term \((z+4)\) from both parts of the equation. This results in: \((z+4)((3z+2) - (z+6))\).

Step 3: Simplify the Remaining Expression

Simplify inside the parentheses by distributing the negative sign in \((3z+2) - (z+6)\). This gives \((3z + 2 - z - 6)\).

Step 4: Combine Like Terms

Combine the terms from the simplified expression: \(3z - z + 2 - 6 = 2z - 4\).

Step 5: Write the Final Factored Expression

Now, substitute back into the expression with the common factor: \((z+4)(2z - 4)\). This is the final factored expression.

Key Concepts

Greatest Common Factor

When we talk about polynomial expressions, the Greatest Common Factor (GCF) refers to the largest polynomial that divides each term of the expression without leaving a remainder. In simpler terms, it's the biggest factor that all terms in the expression can share. This is similar to finding the greatest common divisor in basic arithmetic but applied to algebraic terms.

Consider the expression

• a single variable or number that can be shared by multiple terms (e.g., for numbers like 12, 15, and 21, the GCF is 3)
• a polynomial factor that occurs in each term (e.g., in typical polynomial expression like above).

Finding the GCF is the first step in factoring because it simplifies the process of breaking down more complex expressions. In the provided exercise, the common factor was a polynomial:

• (z + 4)

Finding this factor allowed us to simplify the entire expression easily.

Polynomial Expressions

Polynomial expressions are mathematical phrases that can include variables, coefficients, and operations like addition, subtraction, multiplication, and non-negative integer exponents. An example is a polynomials like:

• (3z+2)
• (z+4)

They can be as simple as a single term, or as complex as a multi-term expression. In our polynomials, each part – (3z+2) and (z+6) – is combined with the common polynomial (z+4).

When working with polynomials, you'll often want to simplify them. This can mean reducing them to fewer terms by combining like terms or factoring them, as was done by identifying common elements and simplifying the expression as a whole.

A critical skill is recognizing these polynomial terms in the context of equations and expressions, making it easier to manipulate and solve them.

Simplification Processes

Simplification processes help make polynomial expressions easier to understand and solve. The goal is to reduce the number of terms and operations, revealing an expression's simplest form. Here’s how it was done in the example exercise:

• Factoring Out: We identified (z+4) as a common factor across multiple terms, which simplifies the complex expression into fewer steps. This reveals the core structure of the expression.
• Distributing and Combining: By redistributing terms inside (3z+2)-(z+6), we simplified to 2z-4. This step involved distributing the negative sign and combining like terms (3z-z and 2-6).
• Finalization: Assembling the simplified expression with the original factor (z+4), we found the final factored form of the original polynomial.

This process is core in algebra to 'clean up' equations, making further operations or solutions less error-prone and more manageable. Understanding simplification can make tough problems simpler and ensure you see the problem's structure clearly.