Problem 31 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ x^{2}-x-2 $$

Step-by-Step Solution

Answer

The trinomial \(x^2 - x - 2\) factors to \((x+1)(x-2)\).

Step 1: Evaluate for a Greatest Common Factor (GCF)

First, we need to check if there is a greatest common factor (GCF) other than 1 that can be factored out from the trinomial \(x^2 - x - 2\). Since the coefficients are 1, -1, and -2, and the GCF of these numbers is 1, there is no GCF other than 1.

Step 2: Identify the Standard Form Factors

Since there is no common factor, we proceed to factor the trinomial in its standard quadratic form. The trinomial \(ax^2 + bx + c\) has \(a = 1\), \(b = -1\), and \(c = -2\). Since \(a = 1\), look for two numbers that multiply to \(c\) (-2) and add to \(b\) (-1).

Step 3: Find Factor Pairs of \(c\)

The pairs of factors for \(-2\) are (1, -2) and (-1, 2). We need to find a pair that sums to \(-1\), which is \(b\).

Step 4: Choose the Correct Factors

The correct pair of numbers is (1, -2) because \(1 + (-2) = -1\).

Step 5: Write the Binomial Factors

Using the pair from the previous step, the trinomial \(x^2 - x - 2\) can be factored into \((x + 1)(x - 2)\).

Step 6: Verify the Factorization

To verify, expand \((x+1)(x-2)\) using the distributive property: \(x(x-2) + 1(x-2)\). That results in \(x^2 - 2x + x - 2\), which simplifies to \(x^2 - x - 2\). The factorization is correct.

Key Concepts

Greatest Common Factor

Understanding the concept of the Greatest Common Factor (GCF) is crucial when factoring trinomials. It's the largest factor that divides all terms in an expression without leaving a remainder. Identifying the GCF effectively reduces the complexity of the equation, making it easier to factor further.
In the trinomial example you're working on, \(x^2 - x - 2\), you'll begin by checking if there's a GCF greater than 1 among the coefficients: 1 (from \(x^2\)), -1 (from \(-x\)), and -2. To find the GCF:

• Identify all the factors for each number.
• 1: just 1
• -1: just 1 and -1
• -2: 1, 2, -1, -2

The only common factor here is 1, meaning there is no GCF other than 1. By confirming this, you can move on to factoring the trinomial without any additional steps in this case. Consider the GCF as the gatekeeper: it simplifies the terms before the main factorization occurs.

Quadratic Trinomials

When dealing with quadratic trinomials like \(x^2 - x - 2\), you're working with an equation in the form \(ax^2 + bx + c\). In this exercise, these coefficients are \(a = 1\), \(b = -1\), and \(c = -2\). The goal in factoring is to express this trinomial as a product of two binomials.
To achieve this, follow these steps:

• Look for two numbers that multiply to give you \(c\) (-2) and add to give you \(b\) (-1).
• Consider the factors of -2: (1, -2) and (-1, 2).
• Among these, (1, -2) adds up to -1, which matches \(b\).

Recognizing these factor pairs is key in setting up your binomial expressions. This process requires some practice, but once familiar, you'll be able to quickly spot these relationships and factor the quadratic trinomials efficiently.

Binomial Factors

Converting a quadratic trinomial into binomial factors allows you to simplify complex problems. For the example \(x^2 - x - 2\), after identifying the factor pair (1, -2), the final step involves expressing the trinomial as a product of two binomials.
The binomial factors here are \((x + 1)\) and \((x - 2)\). To find them, rewrite the numbers found in earlier steps into the binomial structure:

• Place the first number in \((x + number)\).
• Place the second number in \((x - number)\).

Here you get \((x + 1)(x - 2)\). Verifying this involves expanding the binomials using the distributive property (or FOIL method):

• \(x(x-2) + 1(x-2) = x^2 - 2x + x - 2\)
• Simplify to return to the original trinomial \(x^2 - x - 2\).

By understanding how to break down and then reconstruct these equations, you gain a clearer path from the trinomial expression to its binomial factors, reinforcing your algebraic skills.