Problem 7 - Algebra Form and Function

Question

Give the constant term, \(a_{0}\). $$ 15-11 t^{9}-8 t^{4} $$

Step-by-Step Solution

Answer

Answer: The constant term in the given polynomial is \(15\).

Step 1: Identify the constant term

In the given polynomial, look for the term without any variable (such as \(t\)). The polynomial is given by: $$15-11 t^{9}-8 t^{4}$$ In this polynomial, the term without any variable is \(15\).

Step 2: Write down the constant term

The constant term (\(a_{0}\)) in the given polynomial is \(15\).

Key Concepts

Polynomial Expression

A polynomial expression is essentially a combination of variables and coefficients, organized in terms of powers. These powers are non-negative integers, which means you won't see negative exponents in polynomials. Each part of this expression, such as terms like \(15\), \( -11t^9 \), or \( -8t^4 \), represents a separate piece of the overall polynomial puzzle.

When you look at a polynomial, you'll notice it can contain:

• Constants, which are just numbers without any variables or exponents.
• Variables, often represented by letters like \(t\).
• Coefficients, which are the numbers directly in front of the variables (like \(-11\) in \(-11t^9\)).
• Terms, which are the individual parts separated by plus or minus signs.

Degree of Polynomial

Understanding the degree of a polynomial is crucial because it tells you a lot about the expression itself. The degree is defined as the highest power of the variable in the polynomial.

To find it, look for the term with the largest exponent. In this exercise, we have terms like \(15\), \(-11t^9\), and \(-8t^4\). Among these, \(-11t^9\) has the highest exponent, which is \(9\). Therefore, this polynomial is said to be of the 9th degree.

Knowing the degree helps you:

• Understand the shape of the graph if you were to plot it.
• Identify the behavior of the polynomial for very large or very small values.
• Predict the number of roots or solutions it might have.

Constant Term Identification

The constant term in a polynomial is the component that does not change, even when the variables take on different values. It's the term without any variables attached, meaning the power of the variable is zero (\(t^0\)).

In the polynomial given: \(15 - 11t^9 - 8t^4\), the process to find the constant term is straightforward:

• Review each term and look for one without variables. This means checking terms like \(15\), \(-11t^9\), and \(-8t^4\).
• Identify the term that stands alone without variables, which in this example, is \(15\).

Thus, the constant term (denoted as \(a_0\)) in this polynomial is \(15\). Recognizing the constant term is useful, especially when evaluating the polynomial at zero or finding specific values.