Problem 66 - Algebra A Combined Function

Question

Write each polynomial in descending powers of the variable and with no missing powers. See Example 15. $$ 11 z+4 z^{4} $$

Step-by-Step Solution

Answer

The polynomial is \(4z^4 + 0z^3 + 0z^2 + 11z + 0\).

Step 1: Identify the terms of the polynomial

The given polynomial is \(11z + 4z^4\). It consists of two terms: \(11z\) and \(4z^4\).

Step 2: Order terms by degree

To write the polynomial in descending order, we need to order the terms based on the power of the variable \(z\). The term \(4z^4\) has the highest degree, followed by \(11z\), which is equivalent to \(11z^1\).

Step 3: Insert missing powers

In order to ensure no missing powers, we need to include any powers of \(z\) that do not have coefficients in the original polynomial. Thus, we add the terms \(0z^3\), \(0z^2\), and any constant term like \(0\).

Step 4: Write the complete polynomial

Combine all terms, including those with coefficients of 0, to write the polynomial in descending order: \[4z^4 + 0z^3 + 0z^2 + 11z + 0.\]

Key Concepts

Descending Order

When working with polynomials, arranging terms in descending order is a key step. This simply means ordering the terms from the highest power to the lowest power of the variable. In the exercise, the polynomial's terms were initially written as \(11z + 4z^4\). Here, \(4z^4\) needs to be placed first because it has the highest degree, which is 4. Arranging polynomials this way ensures clarity and consistency, making it easier to compare and perform operations on polynomials.

• Scrutinize each term to identify its degree, which is the highest exponent of the variable.
• Ensure that terms with higher degree precede those of lower degree.

This ordering gives a clear hierarchical structure to the polynomial, facilitating better understanding and computation.

Missing Terms

After arranging the terms in descending order, it's crucial to check for missing terms within the polynomial. Missing terms occur when certain powers of the variable are not present because their coefficients are zero, but they need to be included for completeness.For example, the polynomial \(4z^4 + 11z\) was missing terms for \(z^3\), \(z^2\), and the constant term. This is rectified by adding \(0z^3 + 0z^2 + 0\). Including these missing terms serves several purposes:

• Keeps the polynomial structured in a standard form.
• Facilitates easier manipulation during polynomial operations like addition or subtraction.
• Ensures consistency when analyzing the polynomial's structure.

Including all terms, even those with zero coefficients, balances both the aesthetic and functional value of a polynomial expression.

Degrees of Polynomials

A polynomial's degree is a critical concept that defines many of its characteristics. It is determined by the highest power of the variable within the polynomial. Understanding the degree of a polynomial helps in predicting its behavior and the kinds of operations applicable to it.In our expression \(11z + 4z^4\), the highest degree is 4, making it a fourth-degree polynomial. Here are a few points on why degrees are important:

• Defines the leading term of the polynomial, which significantly affects its graph's shape and direction.
• Dictates the number of possible roots or zero crossings of the polynomial on a graph.
• Influences the end behavior of the polynomial when the variable reaches very large or small values.

Understanding the degree gives insight into the polynomial’s complexity and its potential interactions when combined with other polynomials. It's a fundamental aspect when dealing with polynomial functions.