Problem 125 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(x^{3}+3 x^{2}+x+3\)

Step-by-Step Solution

Answer

The expression factors to \((x + 3)(x^2 + 1)\).

Step 1: Group the terms

We start by grouping the terms in pairs to make it easier to factor. The expression is: \[ (x^3 + 3x^2) + (x + 3) \]

Step 2: Factor by grouping

First, factor out the greatest common factor from each group. In \(x^3 + 3x^2\), the common factor is \(x^2\), and in \(x + 3\), there is no common factor other than 1. This gives us: \[ x^2(x + 3) + 1(x + 3) \]

Step 3: Factor out the common binomial factor

Both terms now contain the common factor \((x + 3)\). Factor \((x + 3)\) out: \[ (x + 3)(x^2 + 1) \]

Step 4: Check for further factoring

Now check if \(x^2 + 1\) can be factored further. \(x^2 + 1\) is a sum of squares, which cannot be factored using real numbers. Thus, we leave it as is.

Key Concepts

Factor by Grouping

Factoring by grouping is a method used to simplify polynomials by organizing terms into groups. This method is particularly helpful when dealing with four-term polynomials, such as the expression \(x^3 + 3x^2 + x + 3\). By grouping, you can make the expression easier to manage and factor.

• Start by grouping the terms into pairs. In our example, this means writing it as \((x^3 + 3x^2) + (x + 3)\).
• Next, factor out the greatest common factor (GCF) within each group. For example, in the first group \(x^3 + 3x^2\), the GCF is \(x^2\), while in the second group \(x + 3\), the GCF is 1.
• This process gives us \(x^2(x + 3) + 1(x + 3)\).

The goal of grouping is to find a common binomial factor in the groups, which in this instance is \((x + 3)\). Once you have a common factor, it can be factored out separately. Here, it becomes: \((x + 3)(x^2 + 1)\). Use factor by grouping to simplify expressions effectively, exploring pairs and factoring to reveal those common factors.

Greatest Common Factor

The Greatest Common Factor, or GCF, is the highest number that can divide each term in a polynomial without leaving a remainder. Identifying the GCF is a crucial step in the factoring process and particularly useful in polynomial equations. This method simplifies the expression, making it easier to factor completely.

• You calculate the GCF by looking for the largest factor that is common to all terms in a group.
• In \(x^3 + 3x^2\), the GCF is \(x^2\), because both terms are divisible by \(x^2\).

Finding the GCF lets you extract a common term and simplify the expression further. After factoring out \(x^2\) from \(x^3 + 3x^2\), you get \(x^2(x + 3)\).Always look for the GCF first before using other factoring methods, as it is a simple way to reduce the complexity of a polynomial.

Sum of Squares

Polynomial expressions sometimes involve sums of squares, which are terms like \(a^2 + b^2\). The sum of squares differs from the difference of squares, as it cannot be factored using real numbers.

• Real numbers do not allow for the factoring of sums of squares, such as \((x^2 + 1)\).
• In complex numbers, however, you might express it as a product involving imaginary numbers.

Despite being a part of the expression \((x + 3)(x^2 + 1)\), \(x^2 + 1\) remains as is when factoring with real numbers. Recognizing when a term is a sum of squares is important in determining how far you can simplify an expression with real numbers.Learning about sums of squares helps prevent unnecessary attempts to further factor parts of an expression that cannot be simplified with real numbers. Instead, focus on recognizing such cases and explaining the limitations when working only with reals.