Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The trinomial \(4 a^{2}-5 a-6\) is written in_____ powers of \(a\).

Step-by-Step Solution

Answer

Descending

Step 1: Define Ascending and Descending Powers

In a polynomial, descending powers mean the exponents decrease from left to right, while ascending powers mean the exponents increase from left to right.

Step 2: Check the Exponents Order

Look at the trinomial \(4a^2 - 5a - 6\). The terms are \(4a^2\), \(-5a\), and \(-6\). Here, the exponents are 2, 1, and 0, respectively.

Step 3: Identify the Order

Since the exponents decrease from 2 to 1 to 0 as you read from left to right, the trinomial is written in descending powers.

Key Concepts

Trinomial

A trinomial is a special kind of polynomial. It is a polynomial that has exactly three terms. Each term is separated by a plus (+) or a minus (-) sign, which makes it easy to identify. For example, the trinomial given in the exercise is \(4a^2 - 5a - 6\). This expression consists of three terms: \(4a^2\), \(-5a\), and \(-6\).
It’s helpful to recognize trinomials because they often appear in algebra, especially in problems involving factoring or solving equations. Understanding how the terms interact and how to manipulate them is key in mastering algebraic tasks with trinomials.

Descending Powers

In the world of polynomials and algebra, descending powers refer to the order in which the terms are written. This means the terms are arranged starting from the highest exponent and go down to the lowest exponent as you read from left to right.
For example, in the trinomial \(4a^2 - 5a - 6\), the arrangement follows descending powers. The exponents on the variable \(a\) are 2, 1, and 0, in that order. Thus, this polynomial correctly follows a descending pattern, starting with the term with the highest power down to the constant term with a zero exponent.

• Highest power: \(4a^2\) with exponent 2
• Middle term: \(-5a\) with exponent 1
• Constant term: \(-6\) or \(-6a^0\)

Understanding descending powers helps you properly write and interpret polynomials, making it clear which terms should be a focus during operations like addition, subtraction, or factoring.

Exponents

Exponents are a key part of algebra and polynomials. They tell us how many times a number, known as the base, is multiplied by itself. In the trinomial \(4a^2 - 5a - 6\), each term's exponent on \(a\) offers a glimpse into the term's degree.
Let's break down the exponents in this trinomial:

• The first term \(4a^2\) has an exponent of 2, which means \(a\) is squared (\(a \times a\)).
• The second term \(-5a\) has an exponent of 1, representing \(a\) raised to the first power, which is just \(a\) itself.
• The third term, \(-6\), has no \(a\). It’s a constant term, which can be thought of as \(a^0\). Remember, any number raised to the zero power is 1.
  

Grasping the role of exponents helps in simplifying expressions and solving equations, as they significantly affect how each term adds up or contributes to the polynomial.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It involves everything from solving equations to understanding structures like polynomials, including trinomials.
The trinomial expression in the exercise, \(4a^2 - 5a - 6\), is an excellent example of how algebra uses variables and operations to express relationships. The power of algebra lies in its ability to use these relationships to solve for unknowns, simplify expressions, and model real-world scenarios.

• An equation can set this trinomial equal to zero to solve for \(a\).
• Factoring the trinomial might be a step to further solve or simplify.
• Understanding the structure of algebraic expressions lays the foundation for more advanced mathematics.

Thus, the algebraic manipulation of polynomials is a fundamental skill that facilitates more complex analysis and problem-solving in both academic and practical contexts.