Problem 25 - 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. $$ 2 x^{3}-18 x^{2}+40 x $$

Step-by-Step Solution

Answer

\(2x(x - 4)(x - 5)\)

Step 1: Identify the Greatest Common Factor (GCF)

The given trinomial is \(2x^3 - 18x^2 + 40x\). First, we need to find the greatest common factor of the terms. The coefficients 2, 18, and 40 have a GCF of 2. Each term also contains a factor of \(x\), making the GCF \(2x\).

Step 2: Factor out the GCF

Factor \(2x\) out of each term in the trinomial: \[2x(x^2 - 9x + 20)\]. This leaves us with a simpler trinomial inside the parentheses.

Step 3: Factor the Remaining Trinomial

Now, focus on factoring the trinomial \(x^2 - 9x + 20\). We need two numbers that multiply to 20 and add up to -9. These numbers are -4 and -5.

Step 4: Write the Fully Factored Form

Replace the trinomial with its factors using the numbers found in the previous step: \((x - 4)(x - 5)\). Therefore, the fully factored form of the original expression is \[2x(x - 4)(x - 5)\].

Key Concepts

Greatest Common Factor

In mathematics, the Greatest Common Factor (GCF) is an important aspect when working with algebraic expressions. It represents the largest factor shared by all terms in the expression.
For the trinomial provided, each term is divisible by a shared value, which simplifies the factorization process. In the trinomial \(2x^3 - 18x^2 + 40x\), we start by identifying common factors.

• The coefficients (2, 18, and 40) share a common factor of 2.
• The variable term \(x\) is present in each term, adding it to the GCF.

By determining this, the GCF is found to be \(2x\). Knowing the GCF is key because it is factored out of the original trinomial, setting up the expression for further simplification.

Algebraic Expressions

Algebraic expressions form the basis of many calculations and manipulations in algebra. Understanding how to work with them is crucial for solving problems effectively.
These expressions consist of numbers, variables, and mathematical operators arranged in a specific order.

• Trinomials are a type of algebraic expression with three terms.
• Each term can consist of a coefficient (a number) and a variable raised to an exponent.

In our exercise, \(2x^3 - 18x^2 + 40x\) is a trinomial. Recognizing the structure of such expressions allows us to apply appropriate methods for simplification, such as factoring.

Factorization Process

The factorization process involves breaking down an algebraic expression into simpler components or factors. This is a key skill in algebra that helps in solving equations and simplifying expressions.
The process begins with identifying and extracting the GCF, then continues with factoring the resulting polynomial. Here are the steps:

• The initial step involves determining the GCF, which in our case is \(2x\).
• Factor the GCF out of each term: \(2x(x^2 - 9x + 20)\).
• Focus on the resulting trinomial, \(x^2 - 9x + 20\), and find two numbers that multiply to 20 and add to -9.
• These numbers are -4 and -5, allowing us to express the trinomial as \((x - 4)(x - 5)\).

Once these steps are completed, the expression is completely factored as \(2x(x - 4)(x - 5)\). Understanding these steps is crucial to factor efficiently and effectively.