Problem 23 - College Algebra with Modeling and Visualization

Question

Use grouping to factor the polynomial. \(z^{3}-5 z^{2}+z-5\)

Step-by-Step Solution

Answer

The factored form of the polynomial is \((z^2 + 1)(z - 5)\).

Step 1: Group the polynomial

The polynomial given is \(z^3 - 5z^2 + z - 5\). First, we will attempt to group the terms so that we can factor by grouping. Let's group the first two terms and the last two terms: \((z^3 - 5z^2) + (z - 5)\).

Step 2: Factor out the Greatest Common Factor (GCF) in each group

Now, look at each group separately and factor out the greatest common factor (GCF) from each of them. From the first group \(z^3 - 5z^2\), the GCF is \(z^2\), so we factor it out: \(z^2(z - 5)\). In the second group \(z - 5\), the GCF is 1, and as it is already factored: \(1(z - 5)\).

Step 3: Factor out the common factor

Now, observe that both terms, \(z^2(z - 5) + 1(z - 5)\), have a common factor of \(z - 5\). Factor \(z - 5\) out of both terms: \((z^2 + 1)(z - 5)\).

Step 4: Review the factorization

We have now factored the polynomial \(z^3 - 5z^2 + z - 5\) into \((z^2 + 1)(z - 5)\). This is the final factorization of the given polynomial.

Key Concepts

Grouping Method

The grouping method is a handy tool for factoring polynomials, especially when dealing with four-term polynomials. It involves the strategic arrangement of terms, allowing us to identify common factors. This method simplifies a complex polynomial into more manageable parts.

Here’s how it generally works:

• Divide the polynomial into two groups.
• Look for a common factor in each group.

By rearranging, or "grouping," terms, you oftentimes make the polynomial less intimidating. For instance, the polynomial \(z^3 - 5z^2 + z - 5\) naturally splits into two pairs, leading to the separate groups: \((z^3 - 5z^2)\) and \((z - 5)\). This division helps establish a uniform structure to proceed with factoring. The key in this method is identifying groups where factoring is feasible, allowing us to further simplify the polynomial.

Greatest Common Factor

The greatest common factor (GCF) is a fundamental tool in algebra, especially when factoring polynomials. It refers to the largest factor that divides all terms in a given polynomial.

The process of identifying the GCF in each group is critical. What you want to do is:

• Inspect each group separately.
• Determine the largest factor common to all terms within that group.

For example, in the first group, \(z^3 - 5z^2\), the GCF is \(z^2\). Factoring \(z^2\) out results in \(z^2(z - 5)\). In the second group, \(z - 5\), the GCF is simply 1 since there are no common variables or coefficients to both terms. Even if the GCF is 1, acknowledging it ensures completeness of the factoring process. Finding the GCF streamlines the problem-solving process, particularly in simplifying polynomials.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations. They form the backbone of algebra, allowing us to perform various mathematical operations and manipulations.

An algebraic expression like \(z^3 - 5z^2 + z - 5\) shows complexity due to its variables and powers. To approach such an expression:

• Understand the individual components: terms and their degrees.
• Determine possible factoring techniques, such as using the grouping method.

Expressions can be simplified or factored to aid in solving equations or analyzing patterns. When factoring, we aim to express the polynomial as the product of simpler polynomials. This gives us clarity and often helps in finding roots or zeros of the polynomial. Mastery over handling algebraic expressions paves the way for understanding more advanced mathematical concepts.