Problem 29 - Algebra and Trigonometry

Question

29-34 . Factor the expression by grouping terms. $$ x^{3}+4 x^{2}+x+4 $$

Step-by-Step Solution

Answer

The expression factors to \((x + 4)(x^2 + 1)\).

Step 1: Group Terms

First, identify pairs of terms that can be grouped together. In the expression \(x^3 + 4x^2 + x + 4\), the natural grouping is: \((x^3 + 4x^2) + (x + 4)\).

Step 2: Factor Out Common Factors in Each Group

For the first group \(x^3 + 4x^2\), factor out \(x^2\): \[ x^2(x + 4) \]. For the second group \(x + 4\), the expression is already in factored form \((x + 4) \times 1\).

Step 3: Factor By Grouping

Now that both groups have a common factor \((x + 4)\), factor it out: \[ x^2(x + 4) + 1(x + 4) = (x + 4)(x^2 + 1) \].

Key Concepts

Polynomial Expressions

Polynomial expressions are made up of variables and constants. These are arranged in terms of powers, such as squares or cubes. Each term is called a monomial, and several monomials combine to form a polynomial. In our example, we have four terms: \(x^3 + 4x^2 + x + 4\).
These terms each have different degrees. The degree of a term is the exponent of its variable, and in a polynomial, the term with the highest degree determines its overall degree. Here, \(x^3\) has the highest degree, making our polynomial a cubic expression.
To effectively manage polynomials, understanding their structure is essential. This includes identifying like terms, which are terms that contain the same variables raised to the same powers. Although the given expression in this task has no like terms, being able to spot them is a useful skill for simplifying more complex expressions.

Algebra

Algebra is an area within mathematics that deals with symbols and the rules for manipulating those symbols. These symbols are usually variables representing numbers, and can be used to express general mathematical operations or relationships.
The goal in algebra is often to solve equations by finding values for the unknowns, represented as variables. However, it also involves transforming expressions, such as by factoring. In our problem, understanding the basics of algebra allows us to apply techniques like factoring by grouping. This method leverages the properties of numbers and expressions to simplify or restructure them.

• It allows for breaking down complex problems into simpler parts.
• Facilitates solving equations more efficiently.

Algebra is foundational for solving not only polynomials but a wide variety of problems in math and real-life applications.

Factoring Techniques

Factoring is a crucial technique in algebra used to break down expressions into simpler, multiplied components. This is particularly important when dealing with polynomials, as it can reveal roots or solutions of equations. Factoring techniques like grouping help simplify polynomial expressions.
Factoring by grouping involves several steps. First, the polynomial is divided into groups of terms that have common factors. This might not be straightforward for every expression, but it is very effective here. For instance, we grouped \(x^3 + 4x^2\) and \(x + 4\).

• Identify pairs or groups of terms to factor.
• Extract the common factor from each group.

Once each group is fashioned into a similar binomial, a common factor across these groups can be extracted and rearranged. This is how we arrived at \((x + 4)(x^2 + 1)\) as the factored form of our original expression. It's a method built on recognizing patterns and structures in expressions.