Problem 57 - 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. $$ x^{3}-2 x^{2}-24 x $$

Step-by-Step Solution

Answer

The trinomial factors to \(x(x - 6)(x + 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look at each term in the trinomial \(x^{3}-2x^{2}-24x\) to identify the greatest common factor. The terms are \(x^3, -2x^2, \text{and} -24x\). The GCF here is \(x\) since each term contains an \(x\) and it is the highest power common to all terms.

Step 2: Factor out the GCF

Divide each term in the trinomial by the GCF, \(x\), and factor it out: \[ x(x^2 - 2x - 24) \] This simplifies the expression by removing the common factor.

Step 3: Factor the Simplified Trinomial

Now focus on the quadratic trinomial \(x^2 - 2x - 24\). We need to factor this into two binomials of the form \((x + p)(x + q)\). We need to find two numbers, \(p\) and \(q\), which multiply to \(-24\) (the constant term) and add to \(-2\) (the coefficient of the middle term).

Step 4: Determine the Numbers for Factorization

The numbers that multiply to \(-24\) and add to \(-2\) are \(-6\) and \(4\), since \[-6 imes 4 = -24\] and \(-6 + 4 = -2\).

Step 5: Write the Binomial Expression

Using the numbers determined in Step 4, write the trinomial \(x^2 - 2x - 24\) as: \[ (x - 6)(x + 4) \]. Attach this factorization to the GCF factored out earlier, so the complete factorization of the original expression is: \[ x(x - 6)(x + 4) \].

Key Concepts

Greatest Common Factor

The Greatest Common Factor, or GCF, is the largest number or expression that is a factor of two or more numbers or terms. In algebra, we often deal with polynomials where the GCF is the highest power of a common variable shared by each term.

To find the GCF in the trinomial \(x^3 - 2x^2 - 24x\), we need to look at each term separately:

• \(x^3\)
• \(-2x^2\)
• \(-24x\)

Each term includes the variable \(x\), making \(x\) the common factor across all terms. It is also the highest power of \(x\) that appears in each term. Therefore, \(x\) is the GCF here.

Recognizing and factoring out the GCF simplifies polynomials and reduces the complexity of algebraic expressions, making further factorization more straightforward.

Trinomial

A trinomial is a type of polynomial consisting of three terms. It's generally written in the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

In the original problem, once we factor out the GCF, we are left with the quadratic trinomial \(x^2 - 2x - 24\). Quadratic trinomials are particularly important because they can often be factored into smaller binomial expressions using methods like factoring by grouping or utilizing the quadratic formula.

The goal with trinomials is to simplify them into products of binomials, if possible, which helps in solving equations or simplifying expressions later on. For instance, we transformed the simplified trinomial \(x^2 - 2x - 24\) into \((x - 6)(x + 4)\), facilitating easier manipulation and solution of algebraic problems.

Polynomial

Polynomials are expressions that consist of variables and coefficients, involving operations of addition, subtraction, and multiplication. They can have multiple terms like monomials (one term), binomials (two terms), and trinomials (three terms).

In the expression \(x^3 - 2x^2 - 24x\), we have a polynomial with three terms, each varying in degree. The term \(x^3\) represents the highest degree, which tells us the general nature and behavior of the polynomial function this expression represents. Polynomials can often be simplified or manipulated by factoring.

Understanding the structure of polynomials is key as it guides factorization, which in turn is crucial for simplifying, solving, and graphing algebraic expressions. Each step in the polynomial factorization involves recognizing patterns or applying algebraic identities to break down the polynomial into simpler forms, such as in the factorization process illustrated in the provided solution.