Problem 45 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{8+18 x-5 x^{2}}{10+31 x+15 x^{2}}\)

Step-by-Step Solution

Answer

The expression \(\frac{8+18x-5x^2}{10+31x+15x^2}\) is simplified to \(\frac{(-5x - 2)(x - 4)}{(3x + 2)(5x + 5)}\).

Step 1: Factor the Numerator and Denominator

The numerator is \(8 + 18x - 5x^2\). The proper arrangement is \(-5x^2 + 18x + 8\). We need to factor this quadratic expression. It factors to \((-5x - 2)(x - 4)\).The denominator is \(10 + 31x + 15x^2\). Rewriting it as \(15x^2 + 31x + 10\), it factors to \((3x + 2)(5x + 5)\).

Step 2: Write the Expression in Factored Form

After factoring, the rational expression is written as:\[\frac{(-5x - 2)(x - 4)}{(3x + 2)(5x + 5)}\].

Step 3: Simplify the Rational Expression

Check for common factors in the numerator and the denominator. In this case, there are no common factors. Therefore the expression cannot be simplified further beyond its factored state.

Key Concepts

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of two binomials. This process is crucial for simplifying rational expressions as it allows us to break down complex polynomials into manageable pieces. In our example, the quadratic \(8 + 18x - 5x^2\) is rearranged to \(-5x^2 + 18x + 8\).

• This makes it easier to identify the factors.
• To factor it, look for two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle term's coefficient.

For the numerator, it factors into\((-5x - 2)(x - 4)\), while the denominator is transformed into \(15x^2 + 31x + 10\), which factors as \((3x + 2)(5x + 5)\). Understanding factoring is key to simplifying rational expressions, as it reveals possible cancellations.

Numerator and Denominator

In rational expressions, the numerator and denominator play a crucial role in simplification. The numerator is the top part of the fraction, and the denominator is the bottom part. Simplifying involves factoring both components, and then potentially reducing the expression by cancelling out any common factors. In the exercise provided:

• The numerator \(-5x^2 + 18x + 8\) factors to \((-5x - 2)(x - 4)\).
• The denominator \(15x^2 + 31x + 10\) factors to \((3x + 2)(5x + 5)\).

You first rewrite the expression in its factored form. Understanding the roles of the numerator and denominator is foundational in learning how rational expressions behave and are simplified.

Common Factors

Identifying common factors in the numerator and the denominator of a rational expression is the essential step in simplification. A common factor is a term that appears in both parts of the expression. For example, if both the numerator and the denominator have a factor of \((x + 1)\), they can be cancelled out. But in our current equation, \((-5x - 2)(x - 4)\) over \((3x + 2)(5x + 5)\),

• There are no terms that appear in both the numerator and the denominator.
• This means the expression is already as simplified as it can be.

Recognizing common factors may often allow for simplification, but don't assume they always exist! In cases like this one, the key is ensuring the expression is in its simplest possible form.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator and the denominator. Simplifying them involves factoring these polynomials and using the properties of fractions to reduce the expression when possible. Here’s a simple breakdown:

• Factor both the numerator and denominator into their simplest algebraic expressions.
• Cancel any common factors to simplify the expression further.
• As a result, the expression can sometimes be reduced to a simpler equivalent form.

In the exercise, we see the rational expression starts as \(\frac{8+18x-5x^2}{10+31x+15x^2}\) and factors into \(\frac{(-5x - 2)(x - 4)}{(3x + 2)(5x + 5)}\). Unfortunately, no common factors exist to cancel in this instance, demonstrating that not all rational expressions can be simplified beyond factoring. However, understanding how to manipulate these expressions makes solving complex algebraic equations manageable and can reveal the simplest representation of the equation.