Problem 4 - Elementary and Intermediate Algebra

Question

The polynomial \(2 t^{4}+3 t^{3}-4 t^{2}+5 t-6\) is written in _______ powers of \(t\)

Step-by-Step Solution

Answer

The polynomial is written in descending powers of \(t\).

Step 1: Understanding the Problem

We are given the polynomial \(2t^4 + 3t^3 - 4t^2 + 5t - 6\) and asked to determine how it is organized in terms of the powers of \(t\). We need to identify if the powers of \(t\) are arranged in ascending or descending order.

Step 2: Identify the Powers of \(t\)

The polynomial terms are \(2t^4\), \(3t^3\), \(-4t^2\), \(5t\), and \(-6\). The exponents of \(t\) are 4, 3, 2, 1, and 0, respectively.

Step 3: Determine the Order of Powers

The sequence of exponents is 4, 3, 2, 1, 0. The order of the exponents decreases from the highest value (4) to the lowest (0), which means the powers of \(t\) are in descending order.

Step 4: Conclude the Order of the Polynomial

Since the powers of \(t\) are listed from the largest power to the smallest, the polynomial \(2t^4 + 3t^3 - 4t^2 + 5t - 6\) is written in descending powers of \(t\).

Key Concepts

Exponent order

When dealing with polynomials, understanding how the exponents of variables are arranged gives insight into the structure and properties of the expression. In any polynomial, terms are often organized by their exponents to simplify manipulation and comprehension. For the polynomial \(2t^4 + 3t^3 - 4t^2 + 5t - 6\), the exponents for the variable \(t\) are \(4\), \(3\), \(2\), \(1\), and \(0\).
Exponents tell us the power to which the variable is raised. They impact the polynomial's degree, which is the highest exponent in the polynomial. In our example, the degree is \(4\), because the largest exponent is \(4\). By examining the exponents, one can determine if they are organized in ascending or descending order, affecting how we summarize or simplify our given polynomial.

Descending order

The concept of descending order in a polynomial refers to listing the terms from the highest to the lowest power of the variable. This is the conventional format for writing polynomials, which aids in both understanding the structure and performing operations such as addition and multiplication with ease.
In the polynomial \(2t^4 + 3t^3 - 4t^2 + 5t - 6\), notice that the exponents of \(t\) decrease progressively. They start at \(4\) with \(2t^4\) and reduce down to \(0\) at the constant term \(-6\). This arrangement is a clear indicator that the polynomial is written in descending order. Using descending order makes operations with polynomials, like differentiation and integration, straightforward, as you always begin with the highest power.
The convention of descending order not only supports better comprehension but also aligns with how many mathematical operations and software process polynomials.

Polynomial terms

Polynomials are algebraic expressions made up of terms. Each term consists of a constant coefficient multiplied by a variable raised to a non-negative integer exponent. They can have any number of terms, as long as the exponents are non-negative integers.
In our polynomial \(2t^4 + 3t^3 - 4t^2 + 5t - 6\), you can identify five individual terms:

• \(2t^4\) - the first term with coefficient \(2\) and exponent \(4\)
• \(3t^3\) - the second term with coefficient \(3\) and exponent \(3\)
• \(-4t^2\) - the third term with coefficient \(-4\) and exponent \(2\)
• \(5t\) - the fourth term with coefficient \(5\) and exponent \(1\)
• \(-6\) - the constant term

The structure of these terms and their coefficients determine the characteristics of the polynomial, such as its degree and behavior as inputs change. Each term contributes differently based on its power and coefficient.