Problem 23 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ \frac{1}{2} s t+\frac{3}{2} s t $$

Step-by-Step Solution

Answer

The simplified polynomial is \( 2st \).

Step 1: Identify Identical Terms

The given expression is \( \frac{1}{2} s t + \frac{3}{2} s t \). Both terms, \( \frac{1}{2} s t \) and \( \frac{3}{2} s t \), contain the same variables \( s \) and \( t \) raised to the same power, hence they are like terms.

Step 2: Add Coefficients of Like Terms

Add the coefficients of the like terms: \( \frac{1}{2} \) and \( \frac{3}{2} \). This gives us \( \frac{1}{2} + \frac{3}{2} = \frac{4}{2} \).

Step 3: Simplify the Coefficient

Simplify the fraction \( \frac{4}{2} \). This simplifies to \( 2 \).

Step 4: Write the Polynomial in Descending Powers

With the simplified coefficient, write the polynomial as \( 2 s t \). Here, the polynomial is already in descending order in terms of any variable \( s \) or \( t \) which have the same degree.

Key Concepts

Like Terms

When simplifying polynomials, identifying 'like terms' is essential. Like terms are terms that involve the same variables raised to the same powers. This equivalency allows them to be combined. For instance, in the term \( \frac{1}{2} s t + \frac{3}{2} s t \), both parts contain the variables \( s \) and \( t \), each raised to the first power. Hence, these are like terms. Identifying like terms helps to streamline complex expressions, making them much easier to manipulate and simplify. Since polynomials can include numerous terms, spotting these common variables is a key step in polynomial operations.

Coefficient Addition

In polynomial simplification, after identifying like terms, the next step is coefficient addition. Coefficients are the numerical parts in front of any variables. Adding these together helps simplify the expression further.
For example, in the expression \( \frac{1}{2} s t + \frac{3}{2} s t \), you would add the coefficients directly: \( \frac{1}{2} + \frac{3}{2} = \frac{4}{2} \).
Once added, simplify the resulting number. Here, \( \frac{4}{2} \) simplifies to \( 2 \). Thus, the expression becomes \( 2s t \).

• Always ensure the variables and powers align before adding coefficients.
• This step condenses terms to make handling them easier.

Descending Power Order

Writing a polynomial in descending power order is another essential step. This involves arranging terms based on the powers of a key variable, generally starting from the highest to the lowest. It helps in maintaining a standard form that math users recognize and utilize.

• In our expression \( 2 s t \), deciding on descending order is simple as both the \( s \) and \( t \) have the same powers.
• This enables the polynomial to stay neatly organized.

When expressions grow in complexity, arranging them in descending order makes it easier to see which terms can be further manipulated or simplified.