Problem 24 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ \frac{2}{5} a t+\frac{1}{5} a t $$

Step-by-Step Solution

Answer

\( \frac{3}{5} a t \)

Step 1: Identify Like Terms

Examine the polynomial expression and identify like terms. In this case, both \( \frac{2}{5} a t \) and \( \frac{1}{5} a t \) are like terms since they contain the same variables \( a \) and \( t \) raised to the same powers.

Step 2: Add the Coefficients

Since the terms are like terms, you can combine them by adding their coefficients. Add \( \frac{2}{5} \) and \( \frac{1}{5} \): \[ \frac{2}{5} + \frac{1}{5} = \frac{3}{5} \].

Step 3: Combine the Terms

Multiply the sum of the coefficients by the common variable part \( at \): \[ \frac{3}{5} a t \].

Step 4: Write in Descending Order

Since there is only one term, it is already in descending order. The simplified polynomial is \( \frac{3}{5} a t \).

Key Concepts

Understanding Like Terms

In algebra, simplifying polynomials often involves combining like terms. Like terms are terms that contain the exact same variables, each raised to the same power. For instance, in the polynomial given, both \( \frac{2}{5}at \) and \( \frac{1}{5}at \) are like terms because they share the same variables \( a \) and \( t \), both raised to the power of 1.
Identifying like terms is crucial because it allows us to combine them together, reducing the expression to its simplest form. When tackling polynomials, always take a close look at each term's variable components to spot these similarities.

• Like terms must have identical variable parts.
• The powers of the variables must match.
• This concept allows for easier simplification of expressions.

Cracking Coefficients

Coefficients are the numerical parts of the terms in a polynomial. They are the numbers that multiply the variables. In our example, the terms \( \frac{2}{5}at \) and \( \frac{1}{5}at \) have coefficients \( \frac{2}{5} \) and \( \frac{1}{5} \) respectively.
To simplify the polynomial, we need to add these coefficients. This step is possible because we are dealing with like terms, so only the coefficients combine. When you add \( \frac{2}{5} \) and \( \frac{1}{5} \), it results in \( \frac{3}{5} \). The variable part \( at \) remains unchanged.

• Coefficients tell us how many times the term should be counted.
• Combine coefficients by simple addition if they are associated with like terms.
• This makes simplifying polynomials quick and manageable.

Arranging in Descending Order

Organizing polynomials in descending order involves arranging the terms according to the powers of a chosen variable, usually from highest to lowest. In mathematical terms, it often starts with the term having the highest power and descends from there.
For single-term expressions such as \( \frac{3}{5}at \), there isn't much arranging to do because there's only one term, making it automatically in descending order. If there were multiple terms, we would align them such that the term with the higher power comes first.

• Descending order isn't needed for single-term polynomials, but it's key in multi-term expressions.
• Helps in organizing data for easier reading and solving.
• It is a standardized way to present polynomials consistently.