Problem 29 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{2 x+3}{2 x^{2}-3 x-9} \cdot\left(x^{2}-9\right)$$

Step-by-Step Solution

Answer

The product in the simplest form is \(x+3\).

Step 1: Expand \(x^{2}-9\)

First, the expression \(x^{2}-9\) is a difference of squares. So, it can be expanded to \((x-3)(x+3)\).

Step 2: Multiply the numerators and the denominators

Next, multiply the numerators of both the fractions together. Likewise, multiply the denominators together. This gives: \((2x+3)(x-3)(x+3)\) in the numerator position and \((2x^{2}-3x-9)\) in the denominator.

Step 3: Simplify the expression

At this point, you can now simplify the expression. The term \((2x+3)\) in the numerator position and \(2x^{2}-3x-9\) in the denominator have a common factor, \(x-3\). Remove these from both numerator and denominator positions. After simplifying, the final answer becomes \((x+3)\).

Key Concepts

Difference of Squares

Understanding the concept of a "Difference of Squares" is crucial for simplifying certain algebraic expressions. This pattern occurs when you have two perfect squares separated by a subtraction sign. The general formula to recognize and simplify a difference of squares is:

• If you have an expression like this: \( a^2 - b^2 \)
• You can rewrite it as: \((a-b)(a+b)\)

In this exercise, \( x^2 - 9 \) is a clear example of a difference of squares.
Here, \( x^2 \) and \( 9 \) are the perfect squares (since \( 9 = 3^2 \)). So, you rewrite \( x^2 - 9 \) as \((x - 3)(x + 3)\), which is step one in simplifying our original expression.

Factoring

Factoring is a method used to break down more complex expressions into simpler parts or factors that, when multiplied, will produce the original expression.
To factor effectively, you need to look for common patterns, one of which is a difference of squares as mentioned before. In the provided exercise, after identifying \(x^2 - 9\) as a difference of squares, we factor it into \((x-3)(x+3)\).
Next, you identify common factors across the entire rational expression. In our example, the denominator \(2x^2 - 3x - 9\) can be broken down by finding roots or using methods like the quadratic formula to facilitate factoring. This expression can be transformed to \((x-3)\) \( (\text{Other factors}) \) for simplification purposes.
Once you factor completely, it's simple: remove or "cancel out" common factors that appear in both the numerator and the denominator. Therefore, simplification becomes more manageable as unnecessary parts are removed, leading to your answer.

Rational Expressions

Rational expressions are ratios of two polynomials, much like fractions are ratios of integers. Simplifying rational expressions involves a few key steps:

• First, examine both the numerator and the denominator separately to factor them if possible.
• Next, identify and cancel out any common factors present both in the numerator and denominator.
• Finally, rewrite the simplest form of the expression without these common parts.

In the given problem, we initially have the expression \(\frac{(2x+3)(x-3)(x+3)}{2x^2 - 3x - 9}\).
By cancelling the common factor \((x-3)\) from both parts of the rational expression, we end up with only the components that cannot be reduced further.
Thus, the simplified rational expression is \(x + 3\). Always ensure your final findings are simplified to the fullest to reveal the most concise form of the expression.