Problem 77 - Elementary Algebra

Question

Create your own trinomial of the form \(a x 2+b x+c\) that factors. Share it, along with the solution, on the discussion board.

Step-by-Step Solution

Answer

The trinomial \(x^2 + 5x + 6\) factors to \((x + 2)(x + 3)\).

Step 1: Determine 'a', 'b', and 'c'

Choose values for the coefficients of the trinomial so that it can be factored. Let's take an example where these values are easy to manage. Let \(a = 1\), \(b = 5\), and \(c = 6\).

Step 2: Formulate the Trinomial

Using the values chosen for \(a\), \(b\), and \(c\), create the trinomial. The expression will be: \(x^2 + 5x + 6\).

Step 3: Identify the Factoring Method

Recognize that this is a simple trinomial that can be factored using the method of finding two numbers that multiply to \(c\) (which is 6) and add up to \(b\) (which is 5).

Step 4: Find the Numbers

Identify two numbers that multiply to 6 and add to 5. These numbers are 2 and 3, because \(2 \times 3 = 6\) and \(2 + 3 = 5\).

Step 5: Write the Factored Form

Write the trinomial as a product of two binomials using the identified numbers. Therefore, \(x^2 + 5x + 6\) can be factored as \((x + 2)(x + 3)\).

Key Concepts

Trinomial Expression

A trinomial expression is an algebraic expression composed of three terms. Typically, it's in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients and \(x\) is the variable. Each term in the trinomial represents a different power of the variable, and they're arranged in descending order of degree. These expressions are widespread in algebra and represent quadratic equations once equated to zero.

To understand a trinomial expression practically, consider the example given in the exercise: \(x^2 + 5x + 6\). Here:

• \(a = 1\), which is the coefficient of the quadratic term \(x^2\).
• \(b = 5\), representing the linear term \(x\).
• \(c = 6\), the constant term without a variable.

Understanding the components and structure of a trinomial allows us to apply various algebraic techniques to manipulate or solve the expression effectively.

Binomial Factors

When factorizing a trinomial expression, we aim to rewrite it as the product of two binomial expressions. A binomial is simply a polynomial with two terms, like \(x + 2\) or \(x + 3\).

To factor a trinomial like \(x^2 + 5x + 6\), you're looking for two binomials whose product gives you the original trinomial. This involves identifying two numbers that multiply to the constant term (6 in this case), and add to the linear coefficient (5 here).

In our exercise, these numbers are 2 and 3, since they satisfy both requirements:

• Their product is 6: \(2 \times 3 = 6\).
• Their sum is 5: \(2 + 3 = 5\).

Thus, the trinomial \(x^2 + 5x + 6\) can be expressed as the product of the binomials \((x + 2)\) and \((x + 3)\). This is vital for further simplifications or solving equations where the trinomial is set to zero.

Algebraic Factoring Methods

Factoring is a method of breaking down a complex expression into simpler pieces, which can then be used for further algebraic manipulations. There are several algebraic factoring methods, but when it comes to trinomials of the form \(x^2 + bx + c\), one of the most straightforward methods is by inspection or trial-and-error.

The goal is to find two numbers that both add up to \(b\) and multiply to \(c\). This can often be done through simple mental arithmetic for smaller and simpler trinomials, as seen in the example \(x^2 + 5x + 6\).

Other methods exist too, such as:

• Using the quadratic formula to find roots and then express the trinomial in terms of its factors related to these roots.
• Completing the square when rearranging the expression, which is more often used to solve than to factor directly.

However, for beginners and simple cases, identifying binomial factors by inspection remains a helpful technique. It emphasizes understanding the relationships between coefficients and constants, which is crucial in mastering algebraic factoring.