Problem 52 - 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. $$ x^{2}-4 x-32 $$

Step-by-Step Solution

Answer

The trinomial factors to \((x - 8)(x + 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, we need to check if there is a greatest common factor (GCF) for the trinomial that can be factored out. The terms are \(x^2\), \(-4x\), and \(-32\). The numbers 1, -4, and -32 do not have any common factor other than 1. Therefore, the GCF is 1, and we move to the next step without factoring any GCF out.

Step 2: Setup for Factoring the Trinomial

The trinomial is \(x^2 - 4x - 32\). To factor this trinomial, we look for two numbers whose product is the constant term (-32) and whose sum is the linear coefficient (-4).

Step 3: Find the Pair of Numbers

We need to find two numbers that multiply to -32 and add up to -4. Consider the pairs of factors of 32: \((1, 32), (2, 16), (4, 8)\). Among these, the pair \((4, 8)\) suits our purpose because if one number is negative, their sum is -4. The correct pair is \(-8\) and \(4\).

Step 4: Factor the Trinomial

Rewrite \(-4x\) using the pair of numbers: \(x^2 - 8x + 4x - 32\). Then, use factoring by grouping: \((x^2 - 8x) + (4x - 32)\). Factor each group: \(x(x - 8) + 4(x - 8)\).

Step 5: Complete the Factorization

Notice that \((x - 8)\) is common in both groups, so factor it out: \((x - 8)(x + 4)\). Therefore, \(x^2 - 4x - 32\) factors to \((x - 8)(x + 4)\).

Key Concepts

Greatest Common Factor

The greatest common factor, or the GCF, is a number or expression that divides two or more numbers or terms without a remainder. Finding the GCF is an essential first step when factoring algebraic expressions, such as trinomials.
Before diving into more complex factoring methods, always check if there’s a common factor you can factor out. This simplifies the expression and makes further factoring steps much easier.
For example, consider the trinomial terms: \(x^2\), \(-4x\), and \(-32\). Here, the coefficients are 1, -4, and -32. Upon examining these, the only common factor present among them is 1, which doesn’t change the expressions. This means the trinomial is already simplified concerning the GCF.
If the GCF were other than 1, you’d first factor this out from the entire expression before proceeding with further steps.

Factoring by Grouping

Factoring by grouping is an effective method for factoring trinomials. Once you've ruled out any common factors in the trinomial, look for pairs of terms that can be grouped together to factor out additional expressions from each group.
In our trinomial, \(x^2 - 4x - 32\), we rewrite the linear term \(-4x\) using two numbers that multiply to the constant term \(-32\) but add to the coefficient of the linear term, \(-4\).

• The pair \(-8\) and \(4\) works because \(-8 \times 4 = -32\) and \(-8 + 4 = -4\).

By regrouping:

• We rewrite the expression as \(x^2 - 8x + 4x - 32\).
• Factor out the greatest common factor from each group: \(x(x - 8) + 4(x - 8)\).

Notice how \( (x - 8) \) appears in both groups, allowing you to factor it out, resulting in the factored form: \((x - 8)(x + 4)\).
Combining like terms by grouping simplifies a trinomial and uncovers its factors.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically expressed in the form \(ax^2 + bx + c = 0\). Solving these equations can seem daunting, but understanding their structure is key to factoring them correctly.
The given trinomial, \(x^2 - 4x - 32 = 0\), is an example of a simple quadratic equation where \(a = 1\), \(b = -4\), and \(c = -32\). When dealing with quadratic equations in standard form, it's important to recognize they are often factorable, especially when \(a\) is 1.

• The goal is to express the quadratic as a product of two binomials: \((x + m)(x + n)\).
• Through factoring, we found that \((x - 8)(x + 4)\) satisfies this condition.

Analyzing and factoring quadratic equations help not only in simplifying algebraic expressions but also in solving for the roots efficiently.

Algebraic Expressions

Algebraic expressions, such as trinomials, consist of variables, coefficients, and constants combined using arithmetic operations. Understanding how to manipulate and simplify expressions through factoring is crucial in algebra.
Trinomials, which have three terms, are common in algebraic problems. They can often be broken down into simpler factors, revealing relationships between numbers and making solving for unknown variables easier.

• For the expression \(x^2 - 4x - 32\), we aimed to write it as a product of simpler expressions.
• Factoring allowed us to transform the original trinomial into \((x - 8)(x + 4)\).

Mastering the art of factoring can greatly improve your algebraic skills. It simplifies complex equations and provides insights into their solutions. Always consider different factoring strategies such as recognizing patterns or rearranging them into more familiar forms for ease of manipulation.