Problem 101 - Introductory Algebra for College Students

Question

Factor \(x^{3}+3 x^{2}+2 x\). If \(x\) represents an integer, use the factorization to describe what the trinomial represents.

Step-by-Step Solution

Answer

The fully factored form of the trinomial is \(x(x+1)(x+2)\). If \(x\) is an integer, the trinomial represents the product of three consecutive integers.

Step 1: Identify Common Factors

The given trinomial is \(x^{3}+3 x^{2}+2 x\). The first step in factorisation is always to identify common factors if any exist. Looking at all the terms in the trinomial, it is evident that \(x\) is a common factor.

Step 2: Factor Out the Common Factor

Factor out the common factor \(x\). This gives \(x(x^{2}+3x+2)\). This leaves a quadratic within the parentheses that can further be factored.

Step 3: Factor the Quadratic

Factor the quadratic equation within the brackets. The quadratic can be factored as \((x+1)(x+2)\). This results in the fully factored form of the trinomial as \(x(x+1)(x+2)\).

Step 4: Interpret the Factored Form

If \(x\) represents an integer, each of the factors represents an integer, and the trinomial \(x^{3}+3 x^{2}+2 x\) represents the product of three consecutive integers. In other words, for any integer value of \(x\), the trinomial gives the product of that integer and the next two consecutive integers.

Key Concepts

Common Factor Identification

When facing a polynomial like the one in the exercise, the first step in factorization is to look for a **common factor** across all terms. This is a simple but vital part of simplifying and solving polynomial equations.

Let's dive into what a common factor is. A common factor is a term that is present in each part of the expression. In the case of the polynomial \(x^3 + 3x^2 + 2x\), we identify that \(x\) is a common factor because every term (\(x^3\), \(3x^2\), and \(2x\)) can be divided by \(x\).

*Example:*
- **Expression:** \(x^3 + 3x^2 + 2x\)
- **Common Factor:** \(x\)

Once the common factor is identified, it can be factored out of the equation, simplifying the polynomial for further factorization.

Factoring Quadratics

The process of **factoring quadratics** is a critical skill that follows once you have pulled out any common factors. After removing the common factor from \(x^3 + 3x^2 + 2x\), we are left with \(x(x^2 + 3x + 2)\). Inside this expression is a quadratic: \(x^2 + 3x + 2\).

Factoring quadratics often relies on finding two numbers that both sum up to the linear coefficient (which is \(3\) in this case) and multiply to the constant term (which is \(2\)).

For \(x^2 + 3x + 2\), these two numbers are \(1\) and \(2\).

This allows us to write the quadratic as the product of two binomial expressions:
\(x^2 + 3x + 2 = (x + 1)(x + 2)\).

Using factoring by grouping or identifying patterns (like perfect square trinomials or a difference of squares), these quadratics can easily be split into binomials, simplifying the polynomial further.

Trinomial Factored Form Interpretation

Once you have successfully factored the polynomial, the trinomial is in its **factored form**, which provides valuable insights into the polynomial's behavior and potential solutions.

For the trinomial \(x^3 + 3x^2 + 2x\), the factored form is \(x(x + 1)(x + 2)\). Each of these factors represents a potential zero of the polynomial when set to zero. Thus, the equation \(x(x + 1)(x + 2) = 0\) gives us potential solutions or roots of \(x = 0\), \(x = -1\), and \(x = -2\).

Moreover, this factored form allows us to see the polynomial as the product of three elements. If \(x\) represents an integer, then:

• The first number is \(x\) itself.
• The second is \(x + 1\), the immediate next integer.
• The third is \(x + 2\), the number after that.

So, the entire trinomial represents the product of three consecutive integers when \(x\) is an integer, which can be seen as the product of a triplet of numbers. This interpretation is helpful for real-world applications, such as finding integer sequences or solving integer-based puzzles.