Problem 55 - Elementary and Intermediate Algebra

Question

Write each trinomial in descending powers of one variable and factor. See Example 8 . $$80-24 x+x^{2}$$

Step-by-Step Solution

Answer

The trinomial factors to \((x - 4)(x - 20)\).

Step 1: Rearrange the Trinomial

First, rearrange the terms in the trinomial in descending order of the variable's powers. The given trinomial is \(80 - 24x + x^2\). Rearrange it to \(x^2 - 24x + 80\), with the term \(x^2\) first, followed by \(-24x\), and then the constant \(80\).

Step 2: Set Up for Factoring

To factor the trinomial \(x^2 - 24x + 80\), we need two numbers that multiply to the constant term (80) and add to the coefficient of the middle term (-24).

Step 3: Find Factor Pairs

Consider possible factor pairs of 80 that add up to -24. The possible factor pairs of 80 include (1, 80), (2, 40), (4, 20) etc. Testing for negative sums, the suitable factors are (-4, -20) since \(-4 + (-20) = -24\).

Step 4: Write the Factored Expression

Using the numbers found, we can express the trinomial as a product of two binomials: \((x - 4)(x - 20)\). This is because when you expand \((x - 4)(x - 20)\), it results in \(x^2 - 24x + 80\).

Key Concepts

Descending Powers of a Variable

Descending powers refer to arranging terms of an algebraic expression in order from the highest power to the lowest power of a specific variable. This means starting with the term where the variable is raised to the highest exponent, followed by terms with decreasing exponents.
For example, in the expression \( x^2 - 24x + 80 \), we identify the powers of \( x \):

• \( x^2 \) is the term with the highest power (2).
• \(-24x\) is next, with a power of 1.
• The constant term, 80, technically has a power of 0.

Rearranging terms in descending powers helps in systematically analyzing and working with polynomial expressions, especially when factoring.

Understanding Trinomials

A trinomial is a type of polynomial. It specifically has three terms. These terms are usually separated by plus or minus signs. Let's break down the trinomial step-by-step:

• The prefix 'tri-' suggests three.
• Polynomials include terms with variables raised to different powers, like \( x \).

In our example of \( x^2 - 24x + 80 \), we see three distinct parts:

• 1st part: \( x^2 \)
• 2nd part: \( -24x \)
• 3rd part: constant 80

This trinomial can be factored into two binomials, providing a deeper understanding of how the parts relate.

Algebraic Expression Essentials

Algebraic expressions combine numbers, variables, and arithmetic operations like addition or multiplication. They don't include an "equals" sign. This differentiates them from equations.

For example, in \( 80-24x+x^2 \), we see here a combination of terms without an overall sum or product:

• Numerical coefficients (like 80 or -24)
• Variables (like \( x \)) with and without exponents
• Arithmetic operations (+, -, etc.)

Working with algebraic expressions involves operations such as simplifying, expanding, and factoring to understand the structures of these expressions better.

Transforming into Factored Form

Factored form means writing an expression as a product of its factors. It's a simplified way of expressing a polynomial expression. Factoring is especially useful in solving equations or simplifying expressions.

For example, the process of factoring \( x^2 - 24x + 80 \) involves finding two binomials whose product is the original trinomial. Through factoring, we discovered that:

• The factors are: \( (x - 4)(x - 20) \)
• When expanded, it returns to its original form.

Factoring helps reveal roots or solutions to polynomials and often simplifies algebraic expressions significantly.