Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ r s+4 s t $$

Step-by-Step Solution

Answer

The factored form is \(s(r + 4t)\).

Step 1: Identify Common Factor

First, take a look at the terms in the polynomial \(rs + 4st\). Notice that both terms share a common factor of \(s\). This is the first step in factoring the polynomial.

Step 2: Factor Out the Common Factor

Now that you have identified \(s\) as the common factor, factor it out from each term. Factoring out \(s\), you get: \[s(r + 4t)\]

Step 3: Verify the Factored Expression

Finally, check your work to ensure you've factored correctly. Distribute \(s\) back into the expression to verify:\[s(r + 4t) = sr + 4st\]This confirms that \(s(r + 4t)\) is the correctly factored form of the original polynomial.

Key Concepts

Common Factor

When solving polynomial expressions, it is important to simplify first by identifying any common factors. A common factor in a polynomial is a variable or number that appears in each term of the expression. By removing this factor from each term, we take the first step toward simplifying our polynomial. In the given expression, \( rs + 4st \), both terms have the variable \( s \). This makes \( s \) the common factor. Identifying common factors makes complex polynomials easier to work with.
Here are steps to identify a common factor in any polynomial:

• Look at each term in the polynomial and list all factors.
• Identify variables or numbers that persist across all terms.
• Select one or several common factors that will simplify the polynomial.

Practicing these steps will help you quickly find common factors in any polynomial expression you're working with.

Factored Expression

A factored expression is an essential form of a polynomial where all possible common factors have been removed, and the expression is rewritten in a more simple and manageable form. Once you identify the common factor, you can factor it out, leaving you with a reduced expression that clearly shows the factors involved. In the case of \( rs + 4st \), after factoring out \( s \), the expression becomes \( s(r + 4t) \).
This form helps in:

• Solving equations more easily as it reduces complexity.
• Understanding the structure of the expression for further operations, like division or expansion.

Factored expressions are not only easier to work with but also often reveal insights about the relationships between different parts of the polynomial.

Verify Factored Form

After factoring a polynomial, it is essential to verify that the factored form is correct. Verification ensures that the expression we've derived is truly equivalent to the original. This is done by distributing the factored-out element back through the terms it was taken out from and confirming consistency with the original polynomial.
To Verify:

• Take the factored expression, \( s(r + 4t) \).
• Distribute the \( s \) through the terms in parentheses: \( sr + 4st \).
• Check to see if it results in the original polynomial \( rs + 4st \).

Once verified, we confidently know the polynomial expression is correctly factored, ready for use in solving problems or further manipulations.