Problem 26 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$x^{3}+3 x^{2}-25 x-75$$

Step-by-Step Solution

Answer

The complete factorization of the polynomial \(x^{3}+3 x^{2}-25 x-75\) is \((x+3)(x+5)(x-5)\).

Step 1: Identify the Common Factor

Look at the given polynomial, \(x^{3}+3 x^{2}-25 x-75\), to see if there's a common factor that we can factor out. However, in this case, there's no common factor among all the terms.

Step 2: Rearranging Terms

Sometimes, to make the factorization possible, we should rearrange terms. For this polynomial, we don't need to rearrange since it's already chronologically arranged, i.e., the terms are ordered from highest power to the lowest.

Step 3: Factor by Grouping

Although no common factors among all the terms, there is a common factor in the first pair of terms and a common factor in the second pair of terms. Factor by grouping can be applied here. This method involves two steps: first, group the terms, and second, factor out the common terms in each group. After factoring, it becomes \(x^{2}(x+3) -25(x+3)\). Now, the common factor becomes visible.

Step 4: Factoring out the common factor

Here, \(x+3\) can be factored out since it’s a common factor in both terms, and our polynomial becomes \((x+3)(x^{2}-25)\).

Step 5: Factoring Difference of Squares

The term, \(x^{2}-25\), is the difference of two squares, which can be factored as \((x+5)(x-5)\). Thus, the completely factored polynomial now is \((x+3)(x+5)(x-5)\).

Key Concepts

Factoring by Grouping

Factoring by grouping is a technique used when dealing with polynomials that have four or more terms. Often, direct factoring isn't possible at first glance, but by strategically grouping terms, we can simplify the process. Let's break this down step-by-step using the polynomial \(x^{3}+3x^{2}-25x-75\).

• **Initial Grouping**: Look at the polynomial and try to split it into two pairs: \((x^{3}+3x^{2})\) and \((-25x-75)\).
• **Factoring Each Group**: In the first group, \(x^{3}+3x^{2}\), you can factor out \(x^{2}\), resulting in \(x^{2}(x+3)\). In the second group, \(-25x-75\), \(-25\) can be factored out, giving \(-25(x+3)\).
• **Combine the Factors**: Notice \((x+3)\) in both terms, which means you can factor \((x+3)\) out, resulting in \((x+3)(x^{2}-25)\).

This method simplifies complex polynomials by reducing them into manageable sections, ultimately revealing common factors that weren’t initially obvious.

Difference of Squares

The difference of squares is a special type of factoring used when you have an expression like \(a^{2} - b^{2}\). This can be factored further into \((a+b)(a-b)\). Recognizing this pattern is key to simplifying certain types of polynomials.

In our example, once we factor by grouping and find \((x+3)(x^{2}-25)\), we notice that \(x^{2}-25\) is a difference of squares:

• **Identify Squares**: Here, \(x^{2}\) is \(x\times x\) and \(25\) is \(5\times 5\).
• **Apply Formula**: Applying the difference of squares, \(x^{2}-25\) becomes \((x+5)(x-5)\).

This further simplifies the polynomial into \((x+3)(x+5)(x-5)\). Spotting opportunities to apply the difference of squares significantly reduces the complexity of polynomial manipulation.

Common Factor

Finding a common factor is usually the first step when trying to factor any polynomial. A common factor is a term that divides each term within the expression. Here’s what you should do:

• **Initial Check**: Always begin by examining all the terms in the polynomial for any common factor. A factor common to all is factored out to simplify the expression.
• **Polynomial Example**: In \(x^{3}+3x^{2}-25x-75\), no single term can be divided out evenly across all terms, hence there’s no common factor.
• **Relevance in Overall Process**: Even if you don't find a common factor initially, don’t skip this step. Checking helps guide you to effective methods like grouping or spotting a difference of squares later.

Though in our given expression a common factor was absent, this check is essential practice and sets the stage for employing factorizing techniques like grouping or recognizing patterns.