Problem 64 - Beginning and Intermediate Algebra

Question

Factor completely. $$6 z^{2}+42 z+72$$

Step-by-Step Solution

Answer

The completely factored expression of the given equation is: \[6(z + 3)(z + 4)\]

Step 1: Identify the GCF

First, find the greatest common factor (GCF) of \(6z^2\), \(42z\), and \(72\). The GCF of these terms is \(6\). Divide each term by the GCF: \(6z^2 + 42z + 72 = 6(z^2 + 7z + 12)\)

Step 2: Factor the quadratic expression

Now, focus on factoring the quadratic expression inside the parenthesis: \(z^2 + 7z + 12\). We want to find two numbers that multiply to 12 and add up to 7. We have: 1 * 12 = 12 2 * 6 = 12 3 * 4 = 12 The pair that satisfies the addition is (3, 4). So, we rewrite the middle term as the sum of two terms: \(z^2 + 7z + 12 = z^2 + 3z + 4z + 12\)

Step 3: Factor by grouping

Now, we can factor by grouping. Group the first two terms and the last two terms: \((z^2 + 3z) + (4z + 12)\) Factor out the common factors from each group: \(z(z + 3) + 4(z + 3)\) Now, notice that both terms have a common factor of \((z + 3)\). So, factor out the \((z + 3)\) term: \((z + 3)(z + 4)\)

Step 4: Combine the factors

Finally, combine the GCF and the factored expression from steps 1 and 3: \(6z^2 + 42z + 72 = 6(z^2 + 7z + 12) = 6(z + 3)(z + 4)\) The completely factored expression of the given equation is: \[6(z + 3)(z + 4)\]

Key Concepts

Greatest Common Factor (GCF)

Understanding the concept of the Greatest Common Factor (GCF) is key when factoring polynomials. The GCF is the largest number that divides all given numbers without leaving a remainder. It helps simplify expressions by reducing terms to their simplest form.
When working with polynomials like the one in the exercise: \(6z^2 + 42z + 72\), identifying the GCF is the first step. Here, each term includes a factor of 6, making it the GCF. By factoring out the GCF, the polynomial is simplified to \(6(z^2 + 7z + 12)\).

• This process makes the polynomial easier to work with.
• It reduces the size of the numbers, allowing deeper factoring methods to come into play.

By carefully choosing and factoring out the GCF, more complex expressions become manageable as you tackle quadratic expressions and more.

Quadratic Expression

A quadratic expression is any polynomial where the highest exponent of the variable is 2. It's generally written as \(ax^2 + bx + c\). In our example, this expression is \(z^2 + 7z + 12\) after factoring out the GCF.
Quadratics have a distinctive property: they can often be rewritten as a product of two binomials. This fits into a broader class of equations that look like \((z + n)(z + m)\), provided you choose appropriate numbers for n and m.

• These numbers (n and m) must multiply to give the constant term (here, 12).
• They also need to add up to give the coefficient of the linear term, which is 7 in this case.

Understanding these principles allows you to tackle any quadratic equation by looking for pairs of numbers that meet these conditions. This is crucial when you move towards more advanced algebra.

Factor by Grouping

The technique of "factor by grouping" comes in handy especially when dealing with polynomials that break down into more than simple steps. It's essentially about rearranging and simplifying terms strategically.
Once the GCF is identified and factored out, and the quadratic form is recognized, as in \(z^2 + 7z + 12\), you rewrite it as \(z^2 + 3z + 4z + 12\) to suit grouping. Notice there's an art to selecting these numbers: they must fit our earlier criteria from the quadratic form.

• Group terms into pairs: \((z^2 + 3z)\) and \((4z + 12)\).
• Factor each pair separately: express \(z\) from the first group and \(4\) from the second.
• Each pair reveals a common factor \((z + 3)\).

This method is powerful because it systematically breaks down and simplifies even complex expressions, ultimately leading to clean and clear factors like \((z+3)\) and \((z+4)\), all of which were built on foundational steps.