Problem 24 - Elementary and Intermediate Algebra

Question

Factor. See Example 3 or Example \(10 .\) $$u^{2}+u-42$$

Step-by-Step Solution

Answer

The expression factors to \((u + 7)(u - 6)\).

Step 1: Recognize the quadratic form

The given expression is a quadratic trinomial in the form of \( ax^2 + bx + c \) where \( a=1 \), \( b=1 \), and \( c=-42 \).

Step 2: Identify factors of constant term

Identify two numbers that multiply to \( c = -42 \) and add up to \( b = 1 \). The two numbers are \( 7 \) and \( -6 \) because \( 7 \times (-6) = -42 \) and \( 7 + (-6) = 1 \).

Step 3: Factor the expression

Using the numbers found in Step 2, factor the quadratic expression as \( (u + 7)(u - 6) \).

Step 4: Verify the solution

Expand \( (u + 7)(u - 6) \) to check: \( u^2 - 6u + 7u - 42 = u^2 + u - 42 \), which matches the original expression. Thus, the factorization is correct.

Key Concepts

Quadratic Trinomial

A quadratic trinomial is a type of polynomial that consists of three terms and is of the form \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The presence of the \( x^2 \) (or in our example, \( u^2 \)) term makes it quadratic, distinguishing it from linear expressions.
Quadratic trinomials often appear in algebra problems, and understanding them is vital for solving these problems effectively. In the example \( u^2 + u - 42 \), the coefficients are \( a = 1 \), \( b = 1 \), and \( c = -42 \). These define the structure of the trinomial and guide its factorization.
Recognizing the format allows you to apply methods to simplify and solve algebraic expressions. Understanding the parts of a trinomial helps in identifying the best approach for factoring or solving. It sets a foundation for more complex algebraic manipulations.

Factoring Techniques

Factoring is a process of expressing a polynomial as a product of its factors. There are various techniques, but for quadratic trinomials like the one in our example, a specific approach is used.
To factor \( ax^2 + bx + c \), find two numbers that multiply to \( c \) and add to \( b \). This step is crucial because these numbers help you break down and simplify the trinomial.

• In our example, we need numbers that multiply to \(-42\) and add to \(1\). They are \(7\) and \(-6\).
• Using these, the expression \( u^2 + u - 42 \) can be rewritten as \((u + 7)(u - 6)\).

This method is a basic but powerful tool in algebra, allowing for simplification of expressions for easier manipulation or solving.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operational symbols. They can represent single terms or more complex expressions like our quadratic trinomial.
Understanding how to simplify and manipulate these expressions is central to algebra. Factoring is a key method that helps in breaking down these expressions into simpler, more manageable parts.

• Recognize that expressions like our example \( u^2 + u - 42 \), can be factorized into simpler binomials.
• This makes it easier to find roots or solve equations involving the expression.

Being comfortable with manipulating algebraic expressions allows you to solve equations, simplify complex problems, and apply algebra in various mathematical and real-world contexts.