Problem 64 - Algebra A Combined Function

Question

Write each polynomial in descending powers of the variable and with no missing powers. See Example 15. $$ 6 m^{3}-3 m+4 $$

Step-by-Step Solution

Answer

The polynomial is \(6m^3 + 0m^2 - 3m + 4\).

Step 1: Identify the Terms

The polynomial given is \(6m^3 - 3m + 4\). It consists of three terms: \(6m^3\), \(-3m\), and \(4\).

Step 2: Arrange Terms by Descending Power

To write the polynomial in descending powers of \(m\), we must order the terms starting from the highest power, which is already correctly ordered as \(6m^3 - 3m + 4\).

Step 3: Identify Missing Powers

We notice that the polynomial skips the power of \(m^2\). To include this in the expression, we add \(0m^2\) to the polynomial, resulting in: \(6m^3 + 0m^2 - 3m + 4\).

Step 4: Final Check

Ensure all powers of \(m\) from the highest (3) down to none (0) are included, which they are: \(6m^3 + 0m^2 - 3m + 4\).

Key Concepts

Descending Powers

Writing polynomials in descending powers is a method to arrange the terms of the polynomial starting from the term with the highest power down to the lowest. This is a common convention used in mathematics for both simplicity and consistency. Consider arranging any polynomial by its powers; it helps in operations such as addition, subtraction, and especially differentiation and integration. In our example, we started with the polynomial: - \(6m^3 - 3m + 4\). First, we identify the highest power, which is \(m^3\). We see that the terms are already correctly arranged in descending order:

• The highest power is \(m^3\) with the coefficient 6.
• Next, we look for \(m^2\). It wasn't initially present, but would slot into the sequence before \(m\) if needed.
• Finally, the linear term \(-3m\) and the constant \(+4\).

This arrangement makes it easier to compare polynomial degrees and perform further mathematical operations.

Missing Powers

When a polynomial seems to skip any powers of the variable, these powers aren't missing but often just have a coefficient of zero. Incorporating them explicitly can sometimes simplify certain algebraic manipulations or clarifications. For our polynomial, \(6m^3 - 3m + 4\), we identify a missing power:

• The polynomial does not initially contain any term with \(m^2\).
• We can treat this as a "missing power" and add \(0m^2\) into the expression.

By writing the polynomial as \(6m^3 + 0m^2 - 3m + 4\), you include all powers from the highest down to zero. Doing so can sometimes help visualize the structure of the polynomial or simplify steps in polynomial operations such as synthetic division.

Algebraic Terms

Algebraic terms are the building blocks of polynomials, made up of coefficients and variables raised to a power. Each term in a polynomial is separated by a plus \((+)\) or minus \((-\)) sign. In our polynomial example, we identified these terms as follows:

• \(6m^3\): This is a cubic term. The coefficient is 6, and the variable \(m\) is raised to the 3rd power.
• \(-3m\): This is a linear term. The coefficient is \(-3\), and the power of \(m\) is 1.
• \(4\): This is a constant term, with no variable part, essentially \(m^0\).

Recognizing and outlining each term helps when performing polynomial operations such as addition, subtraction, or multiplication. Properly identifying algebraic terms ensures precision when manipulating polynomials in further exercises or calculations.