Problem 41 - 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. $$ x^{2}-18 x-144 $$

Step-by-Step Solution

Answer

The trinomial \(x^2 - 18x - 144\) factors to \((x+6)(x-24)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, look for the greatest common factor (GCF) of the terms in the trinomial. The expression is \(x^2 - 18x - 144\). Note that all terms are integers and do not have a common factor other than 1, so the GCF is 1.

Step 2: Determine Factorability

Since the GCF is 1, we need to determine if the trinomial can be factored into two binomials. Look for two numbers that multiply to the constant term (-144) and add to the coefficient of the middle term (-18).

Step 3: Find Numbers to Factor By

The two numbers that multiply to -144 and add to -18 are 6 and -24. This is because \(6 \times (-24) = -144\) and \(6 + (-24) = -18\).

Step 4: Write As a Product of Binomials

Using the numbers found in Step 3, factor the trinomial \(x^2 - 18x - 144\) into the product of two binomials: \((x + 6)(x - 24)\).

Step 5: Verify the Factorization

Expand \((x+6)(x-24)\) to check: \((x+6)(x-24) = x(x-24) + 6(x-24) = x^2 - 24x + 6x - 144 = x^2 - 18x - 144\). Since it matches the original expression, the factorization is verified.

Key Concepts

Greatest Common Factor

Before diving into factoring a trinomial, it is crucial to check if there is a Greatest Common Factor (GCF) among the terms. The GCF is the largest number or expression that evenly divides all terms in the polynomial. However, in expressions like \(x^2 - 18x - 144\), you might find, as in this case, the GCF is 1. This means there are no simplifiable common terms other than 1. Identifying the GCF helps simplify the problem and paves the way for smooth factorization. It is a vital first step when tackling any polynomial as it could potentially make the factorization process easier. Remember, taking out the GCF, if it's greater than 1, will result in smaller, more manageable terms to work with.

Binomial Factorization

After determining the GCF, or concluding it is 1, we turn to binomial factorization. This involves rewriting a trinomial as the product of two binomials. To begin with binomial factorization, the strategy is to find two numbers that multiply to the constant term of the trinomial and add to the coefficient of the linear term. For our trinomial, \(-144\) is the constant and \(-18\) is the linear coefficient.

The numbers 6 and -24 fit these requirements: they multiply to \(-144\) and add to \(-18\). With these numbers, the trinomial \(x^2 - 18x - 144\) can be broken down into the binomials \((x + 6)(x - 24)\). Remember, this process involves a bit of trial and error when finding the fitting pair of numbers, but practice makes it easier over time.

Quadratic Expression

A quadratic expression is any polynomial expression of degree two, typically written in the form \(ax^2 + bx + c\). Our example, \(x^2 - 18x - 144\), fits this structure, where:

• \(a = 1\)
• \(b = -18\)
• \(c = -144\)

Quadratics can form parabolas when graphed and are a fundamental concept in algebra. Understanding how to factor these expressions is crucial as it reveals the roots of the equation \(ax^2 + bx + c = 0\). In this case, the roots can be found by setting each binomial factor to zero, leading to \(x+6=0\) and \(x-24=0\). This means \(x = -6\) and \(x = 24\) are the solutions. Quadratic expressions are everywhere in mathematics, making mastery of factoring them a valuable skill.

Problem-solving Steps

Solving trinomials involves a series of clear steps. Here is a systematic approach:

• First, identify the GCF and factor it out. If the GCF is 1, like in our case, we proceed to the next step.
• Next, assess whether the trinomial is factorable into binomials. Look for two numbers that multiply to the constant term and sum to the middle term's coefficient.
• Find these numbers and use them to write the trinomial as two binomials.
• Finally, always verify your work. Expand the binomials to ensure they simplify back to the original trinomial.

Following these problem-solving steps will make sure you're thorough and accurate in your approach. Once you've practiced these steps frequently, factoring trinomials will become intuitive.