Problem 41 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{10 a^{2}+a-3}{15 a^{2}+4 a-3}$$

Step-by-Step Solution

Answer

\(\frac{2a - 1}{3a - 1}\)

Step 1: Identify Patterns for Factoring

Notice that both the numerator and the denominator of the algebraic fraction are trinomials. We can attempt to factor each trinomial separately, as they might be factorable into simpler expressions multiplied together.

Step 2: Factor the Numerator

The numerator is given by \(10a^2 + a - 3\). To factor this trinomial, we will look for two numbers that multiply to \(10 \times -3 = -30\) and add to \(1\). These numbers are \(6\) and \(-5\). Therefore, we can rewrite \(a\) as \(6a - 5a\) and proceed to factor by grouping: \[10a^2 + a - 3 = 10a^2 + 6a - 5a - 3\] \[= (10a^2 + 6a) + (-5a - 3)\] \[= 2a(5a + 3) - 1(5a + 3)\] \[= (2a - 1)(5a + 3)\].

Step 3: Factor the Denominator

The denominator is \(15a^2 + 4a - 3\). Similarly, we find two numbers that multiply to \(15 \times -3 = -45\) and add to \(4\). These numbers are \(9\) and \(-5\). We rewrite \(4a\) as \(9a - 5a\):\[15a^2 + 4a - 3 = 15a^2 + 9a - 5a - 3\] \[= (15a^2 + 9a) + (-5a - 3)\] \[= 3a(5a + 3) - 1(5a + 3)\] \[= (3a - 1)(5a + 3)\].

Step 4: Simplify the Expression

We now replace the factored forms of the numerator and denominator into the original fraction:\[\frac{(2a - 1)(5a + 3)}{(3a - 1)(5a + 3)}\].Notice that \((5a + 3)\) appears in both the numerator and the denominator, so we can cancel this common factor: \[\frac{2a - 1}{3a - 1}.\]

Step 5: Write the Simplified Expression

After simplifying by cancelling the common factor \((5a + 3)\), the simplified expression of the algebraic fraction is \(\frac{2a - 1}{3a - 1}\).

Key Concepts

Factoring Trinomials

When dealing with algebraic fractions, one of the first steps is often factoring trinomials. A trinomial is a polynomial with three terms, generally written in the form \(ax^2 + bx + c\).
In our example, both the numerator and denominator are trinomials. To factor them, we aim to break them down into simpler binomials (two-term expressions) or factors. This process is also known as "factoring by grouping."
To start factoring a trinomial, follow these steps:

• Identify two numbers that multiply to give you the product of \(a\) and \(c\) (the first and last coefficient) and add to \(b\) (the middle coefficient).
• Rewrite the trinomial splitting the middle term using your two numbers.
• Group the terms into two pairs and factor each pair separately.
• Look for a common binomial factor between the pairs and factor it out.

In the given exercise, for the numerator \(10a^2 + a - 3\), we found that the numbers \(6\) and \(-5\) work, while for the denominator \(15a^2 + 4a - 3\), the numbers are \(9\) and \(-5\). These numbers allow us to factor both trinomials into binomial products. Understanding this step forms the foundation for further simplification.

Simplifying Expressions

After factoring the trinomials, our next goal is to simplify the expressions in our algebraic fraction. Simplifying an expression means reducing it to its simplest form, which makes it easier to work with or interpret.
In our example, once the trinomials are factored, we have:

• Numerator: \((2a - 1)(5a + 3)\)
• Denominator: \((3a - 1)(5a + 3)\)

Notice that both the numerator and the denominator share a common binomial factor \((5a + 3)\). This presents an opportunity to simplify the expression by cancelling out this common factor, which is a critical step in the simplification process.
The outcome of this simplification is a simpler fraction that retains the same mathematical properties as the original. Simplifying expressions is not only helpful in reducing complexity but also crucial in presenting solutions in their most elegant form.

Cancelling Common Factors

Cancelling common factors is a fundamental skill in simplifying algebraic fractions. It involves identifying and removing identical factors from the numerator and the denominator.
When a factor appears in both the numerator and the denominator of a fraction, it can be "cancelled" out, effectively reducing the fraction. However, it is important to note that cancellation is only possible if the factor is multiplied by the rest of the expression (not added or subtracted).
In the exercise example, we factored the expressions and found \((5a + 3)\) in both the numerator and the denominator:

• Numerator: \((2a - 1)(5a + 3)\)
• Denominator: \((3a - 1)(5a + 3)\)

Since \((5a + 3)\) is common in both, we can cancel it out, yielding the simpler expression \(\frac{2a - 1}{3a - 1}\).
This kind of reduction simplifies computation and leads to cleaner results. Cancelling common factors ensures that we are left with the simplest possible form of an expression or a fraction. It’s an essential skill not just in algebra, but in any field that requires precise mathematical simplification.