Problem 19 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 2 x^{2}-7 x+5 $$

Step-by-Step Solution

Answer

\((2x - 5)(x - 1)\)

Step 1: Multiply and Rearrange Terms

We start by multiplying the leading coefficient with the constant term. Here, the leading coefficient is 2 and the constant term is 5. Multiply them to get \( 2 \times 5 = 10 \). We need to find two numbers that multiply to 10 and add up to the middle coefficient, which is -7.

Step 2: Find Two Numbers

The two numbers that multiply to 10 and add up to -7 are -5 and -2. Hence, we will rewrite the middle term -7x as -5x - 2x.

Step 3: Rewrite the Trinomial

Using the numbers from Step 2, rewrite the expression \(2x^2 - 7x + 5\) as \(2x^2 - 5x - 2x + 5\).

Step 4: Group Terms

Group the terms in pairs: \((2x^2 - 5x)\) and \((-2x + 5)\).

Step 5: Factor Each Group

Factor each group separately. In \(2x^2 - 5x\), the common factor is x, giving \(x(2x - 5)\). In \(-2x + 5\), the common factor is -1, giving \(-1(2x - 5)\).

Step 6: Factor by Grouping

Since both groups contain \(2x - 5\), factor this common factor out: \((2x - 5)(x - 1)\).

Key Concepts

Understanding Trinomials

A trinomial is a specific type of polynomial. As the name suggests, it consists of three distinct terms. Each of these terms can be a combination of coefficients and variable expressions. For instance, in the expression \(2x^2 - 7x + 5\), the trinomial consists of:

• \(2x^2\) - the quadratic term
• \(-7x\) - the linear term
• \(5\) - the constant term

These terms are usually organized from the highest to the lowest power of the variable \(x\). Understanding the structure of a trinomial is essential, as it helps in further processes such as factoring. In algebra, trinomials are often encountered, especially when dealing with quadratic expressions, which can describe curves like parabolas.

Factoring: Breaking Down Trinomials

Factoring is a crucial skill in algebra that involves breaking down more complex expressions into simpler ones, which when multiplied together yield the original expression. When dealing with trinomials, factoring can simplify expression handling, solving equations, and graphing. The goal is to express the trinomial in a product format, often by using techniques like grouping.

In our exercise, \(2x^2 - 7x + 5\), the process of factoring allowed us to reach the simpler form \((2x - 5)(x - 1)\). Each new factor reveals important roots or solutions to related equations. This is key to solving quadratic equations by setting each factor equal to zero and solving for \(x\).

Polynomials: More than Just Trinomials

Polynomials are expressions made up of variables raised to various powers and multiplied by coefficients.

They include a wide array of expressions, from very simple ones like \(x + 1\) to more complex, multi-term expressions such as the trinomial in our session, \(2x^2 - 7x + 5\). Polynomials are essential building blocks in algebra.

• Monomials, which have just one term (e.g., \(3x\))
• Binomials, which consist of two terms (e.g., \(x - 4\))
• Trinomials, made up of three terms

Working with polynomials involves various operations, such as addition, subtraction, and for our purposes, factoring. Recognizing the form and type of a polynomial, such as whether it is a trinomial, is the first step in choosing the correct method to manipulate it.

Algebraic Expressions and Their Manipulation

Algebraic expressions are combinations of numbers, variables, and operations like addition or multiplication. These expressions form the core of algebra.

The key to mastering algebra is learning how to manipulate these expressions effectively. Manipulation includes tasks like factoring, expanding, or combining like terms. Our task with the trinomial involves rewriting it in a factored form, making it easier to solve equations or understand its properties.

Every process, like the grouping method used for factoring, is a tool for simplifying complex algebraic expressions. This simplification helps not only in practical tasks, such as solving for \(x\), but also in developing a deeper understanding of mathematical relationships.