Problem 57 - Algebra for College Students

Question

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{2 s t-30-12 s+5 t}{3 s t-6-18 s+t}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2s+5}{3s+1}\).

Step 1: Identify and Group Terms

First, we identify terms in the numerator and denominator that can be grouped together for factoring. The expression is \( \frac{2st - 30 - 12s + 5t}{3st - 6 - 18s + t} \). We can group the terms as follows:Numerator: \((2st - 12s) + (5t - 30)\)Denominator: \((3st - 18s) + (t - 6)\).

Step 2: Factor Each Group Separately

Now, in each group, we factor out the greatest common factor.Numerator:\[2st - 12s = 2s(t - 6)\]\[5t - 30 = 5(t - 6)\]Combine to get: \[2s(t-6) + 5(t-6)\]Denominator:\[3st - 18s = 3s(t - 6)\]\[t - 6 = 1(t - 6)\]Combine to get: \[3s(t-6) + 1(t-6)\].

Step 3: Factor by Grouping

Notice that both the numerator and the denominator have a common binomial factor \((t-6)\).Numerator: \[2s(t-6) + 5(t-6) = (t-6)(2s + 5)\]Denominator: \[3s(t-6) + 1(t-6) = (t-6)(3s + 1)\].

Step 4: Simplify the Expression

Simplify the rational expression by canceling the common factor \((t-6)\):\[\frac{(t-6)(2s+5)}{(t-6)(3s+1)} = \frac{2s+5}{3s+1}\].

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used to simplify algebraic expressions, particularly when dealing with complex polynomials. It involves rearranging and grouping terms to reveal common factors.
Here's how it works:

• Identify groups: Split the polynomial expression into pairs of terms that have common factors. This step requires practice to see which terms naturally group together, as shown in the original exercise where terms like \((2st - 12s) + (5t - 30)\) are grouped.
• Factor out common factors: Once grouped, factor out the greatest common factor from each pair. In our example, you factor out \(2s\) from \((2st - 12s)\) and \(5\) from \((5t - 30)\). This helps to simplify the expression significantly.
• Look for a common binomial: After factoring, identify if there is a common binomial in each group. Here, both grouped terms have been simplified to involve \((t-6)\).

When done correctly, factoring by grouping allows the problematic expression to be rewritten in a cleaner, often more manageable form. It is an essential algebraic skill for simplifying rational expressions, like in our example.

Polynomial Simplification

Polynomial simplification involves reducing the complexity of a polynomial by applying algebraic techniques. It simplifies lengthy expressions without altering their underlying values. The key steps include:

• Identify like terms: Like terms have the same variables raised to the same power. Group these together to make simplification easier. In our case, terms sharing common factors are grouped and factored for efficiency.
• Factor completely: Breaking down each polynomial into its simplest factors often reveals patterns. With our expression, both the numerator and denominator were factored to show \((t-6)\), helping in the cancellation process.
• Cancel common factors: This is crucial in simplifying rational expressions. Here, since \((t-6)\) appears in both the numerator and denominator, it is canceled out, simplifying the expression to \(\frac{2s+5}{3s+1}\).

Simplifying polynomials makes them easier to understand and work with, and it's a skill that’s especially valuable in solving complex algebraic problems.

Algebraic Fractions

Algebraic fractions, also known as rational expressions, are fractions in which the numerator and/or the denominator are polynomials. Simplifying algebraic fractions is a fundamental skill in algebra.
Here's how to approach simplifying them:

• Factor both polynomials: The first step is to factor both the numerator and the denominator separately. In our exercise, we saw terms broken down through consistent factoring by grouping.
• Identify and cancel common factors: Once the expression is factored, look for terms that appear in both the numerator and the denominator. These common factors can be canceled out, which we did with \((t-6)\) in our original problem.
• Simplify to the lowest terms: After cancelation, you are left with the simplified form of the algebraic fraction. Our worked example was reduced to \(\frac{2s+5}{3s+1}\) after canceling.

Understanding how to handle algebraic fractions effectively paves the way for more advanced mathematical explorations and applications. This includes operations like addition, subtraction, multiplication, and division of rational expressions.