Problem 8 - Algebra Form and Function

Question

Give the constant term, \(a_{0}\). $$ 7 t^{3}+2 t^{2}+5 t $$

Step-by-Step Solution

Answer

Answer: \(a_{0} = 0\)

Step 1: Identify the constant term

In the given polynomial function, the constant term is the term without any variable (\(t\)) factors. This means that we are looking for a term that has no \(t\)s.

Step 2: Check each term of the polynomial function

The given polynomial function has three terms: \(7t^{3}\), \(2t^{2}\), and \(5t\). Notice that all of these terms contain the variable \(t\). This means that the constant term \(a_{0}\) is \(0\).

Step 3: Write the final answer

Since there is no constant term in the given polynomial function, the constant term \(a_{0}\) is: $$ a_{0} = 0 $$

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients. They involve terms that are combined using addition, subtraction, multiplication, and non-negative integer exponents on variables. The general form of a polynomial in a single variable, say, from the variable \( t \), is expressed as:\[P(t) = a_n t^n + a_{n-1} t^{n-1} + \cdots + a_1 t + a_0\]Here, \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients, and \( t \) is the variable. Each term in the polynomial is composed of a constant multiplied by a power of the variable. The highest power of the variable in the polynomial indicates its degree.- **For example:** In the polynomial \( 7t^3 + 2t^2 + 5t \), the terms \( 7t^3 \), \( 2t^2 \), and \( 5t \) are each composed of a coefficient and a power of \( t \). The degree of this polynomial is 3, as the highest exponent of \( t \) is 3.Understanding polynomials is foundational in algebra, as they appear in various mathematical contexts and real-world situations.

Constant Terms

A constant term in a polynomial is a term that does not contain any variables or variable factors. It is simply a numerical value, sometimes referred to as the "free term." In the polynomial form, it is represented as \( a_0 \), which stands alone without any variable next to it.- **Why it matters:** The constant term is important because it can provide information about the polynomial's value when the variable equals zero. This is because all terms involving variables will evaluate to zero, leaving just the constant term.- **Example:** Consider \( 5x^3 + 2x^2 + 4 \). Here, \( 4 \) is the constant term since it is the part of the polynomial without the variable \( x \).In some cases, such as the polynomial \( 7t^3 + 2t^2 + 5t \) from our exercise, there can be no constant term. This happens when each term in the polynomial product contains at least one variable. As a result, the constant term is zero, signifying that there is no standalone constant in the polynomial.

Variable Identification

Identifying variables in a polynomial is a crucial step in analyzing and solving polynomial expressions. A variable is typically represented by symbols such as \( x \), \( y \), or \( t \), and are placeholders for potential numerical values.- **In polynomials:** Each term is usually composed of a coefficient and one or more variables raised to an exponent. The role of the variable is to change the value of the polynomial when substituted with numbers.- **Identification process:** To identify a variable in a polynomial, look for any uppercase or lowercase letters in the expression. These are the indicators of variables present in each term.- **Example:** In the polynomial \( 7t^3 + 2t^2 + 5t \), the variable is \( t \), and it appears in each term, defining their structure and magnitude.Recognizing variables and understanding their significance in polynomials is foundational for deeper mathematical exploration, as it allows for variable manipulation, simplification, and solving equations derived from these polynomials.