Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the rational expressions. \(\frac{a^{2}+9 a+18}{a^{2}+3 a-18}\)

Step-by-Step Solution

Answer

\(\frac{a + 3}{a - 3}\)

Step 1: Factor the Numerator

The numerator of the rational expression is \(a^2 + 9a + 18\). To factor this expression, find two numbers that multiply to 18 (the constant term) and add to 9 (the coefficient of the linear term). These numbers are 3 and 6. Therefore, the expression can be factored as \((a + 3)(a + 6)\).

Step 2: Factor the Denominator

The denominator of the rational expression is \(a^2 + 3a - 18\). To factor this, find two numbers that multiply to -18 and add to 3. These numbers are 6 and -3. Therefore, the expression can be factored as \((a + 6)(a - 3)\).

Step 3: Simplify the Expression

After factoring both the numerator and the denominator, the rational expression is \(\frac{(a + 3)(a + 6)}{(a + 6)(a - 3)}\). Cancel out the common factor \((a + 6)\) in both the numerator and the denominator, leaving \(\frac{a + 3}{a - 3}\).

Step 4: State the Simplified Expression

The simplified form of the rational expression is \(\frac{a + 3}{a - 3}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the pieces of a puzzle. Each polynomial can often be broken down into simpler parts. These simpler parts, when multiplied together, give back the original polynomial. This step, known as factoring, is crucial when simplifying rational expressions. Take, for example, the polynomial in the numerator of our rational expression: \(a^2 + 9a + 18\). The goal here is to express it as a product of two binomials. To do this, look for two numbers that multiply to 18, the constant term, and add to 9, the coefficient of the middle term.

• Multiply: 3 and 6 give 18.
• Add: 3 plus 6 equals 9.

Thus, you can write the numerator as \((a + 3)(a + 6)\).
Similarly, the denominator \(a^2 + 3a -18\) needs factoring. Find numbers that multiply to -18 and add up to 3.

• Multiply: 6 and -3 produce -18.
• Add: 6 minus 3 equals 3.

Thus, express the denominator as \((a + 6)(a - 3)\). Successfully factoring these polynomials makes simplifying much easier.

Numerator and Denominator

The terms numerator and denominator refer to the top and bottom parts of a fraction, respectively. In a rational expression, the numerator and the denominator are polynomials. Knowing which part is which helps in the simplification process as both parts are treated differently.

• Numerator: Represents the part above the fraction line. For our expression, it's \(a^2 + 9a + 18\).
• Denominator: Represents the part below the fraction line. For the problem at hand, this is \(a^2 + 3a - 18\).

When you factor both parts, you're essentially preparing them for simplification. The goal is to identify any common factors in both the numerator and the denominator, which can then be cancelled out to simplify the expression. This clear separation of roles plays a critical part in understanding how rational expressions work.

Simplification Process

After factoring, simplifying a rational expression involves cancelling out common factors from the numerator and the denominator. This is an essential step, as it reduces the expression to its simplest form.Let's simplify: Take the factored expression \(\frac{(a + 3)(a + 6)}{(a + 6)(a - 3)}\). Here, \((a + 6)\) is a common factor in both the numerator and the denominator. By cancelling \((a + 6)\) out, the expression simplifies to \(\frac{a + 3}{a - 3}\).

• Identify the same factors in both parts of the expression.
• Cancel those factors, simplifying the expression.

Remember that simplification doesn’t change the value of the expression; it merely presents it in a more manageable form. Always check to ensure no further factors in common exist. This process makes even complex rational expressions look neat and tidy.