Problem 20 - College Algebra

Question

Multiply the rational expressions and express the product in simplest form. $$ \frac{6 x^{2}-5 x-50}{15 x^{2}-44 x-20} \cdot \frac{20 x^{2}-7 x-6}{2 x^{2}+9 x+10} $$

Step-by-Step Solution

Answer

The product is expressed in its simplest form as \(\frac{(3x + 10)(4x + 3)(2x - 5)(5x - 2)}{(5x + 2)(3x - 10)(2x + 5)(x + 2)}\).

Step 1: Factor the Numerators and Denominators

First, identify the numerators and denominators of each rational expression and factor them. For \(6x^2 - 5x - 50\), identify two numbers that multiply to \(-300\) (as you get from \(6 \times -50\)) and add to \(-5\). Those numbers are \(-20\) and \(15\). Use these to factor the expression into:\(6x^2 - 5x - 50 = (3x + 10)(2x - 5)\).For \(15x^2 - 44x - 20\), factor into:\( (5x + 2)(3x - 10) \).For \(20x^2 - 7x - 6\), factor into:\((4x + 3)(5x - 2)\).For \(2x^2 + 9x + 10\), factor into:\( (2x + 5)(x + 2) \).

Step 2: Set Up the Factored Expression

Substitute the factored forms into the expression:\[\frac{(3x + 10)(2x - 5)}{(5x + 2)(3x - 10)} \cdot \frac{(4x + 3)(5x - 2)}{(2x + 5)(x + 2)}\]

Step 3: Cancel Common Factors

Now, look for common factors in the numerators and denominators that can be cancelled: - There are no common factors in the numerators and denominators to directly cancel here. Thus, no terms are cancelled, and hence the expression is in its simplest form.

Step 4: Multiply the Remaining Factors

Multiply the remaining factors after cancellation to get the expression:Numerator: \[(3x + 10)(2x - 5)(4x + 3)(5x - 2)\]Denominator: \[(5x + 2)(3x - 10)(2x + 5)(x + 2)\]This gives the final expression in factor form:\[\frac{(3x + 10)(4x + 3)(2x - 5)(5x - 2)}{(5x + 2)(3x - 10)(2x + 5)(x + 2)}\]

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of rewriting a polynomial as a product of other simpler polynomials. It is like decomposing a sentence into individual words. Each polynomial can be broken down until it cannot be factored further. To factor a polynomial, follow these steps:

• Determine the product of the leading coefficient and the constant term.
• Find two numbers that multiply to this product and add up to the middle coefficient.
• Split the middle term using these two numbers and then factor by grouping.

Let's consider the expression \(6x^2 - 5x - 50\). Start by multiplying the leading coefficient (6) by the constant term (-50), which equals -300. Next, find two numbers that multiply to -300 and add to -5, which are -20 and 15. Thus, rewrite and factor the expression: \(6x^2 - 5x - 50 = (3x + 10)(2x - 5)\).
Repeating this process for each polynomial is crucial as it allows you to simplify the rational expressions later and understand their behavior.

Multiplication of Rational Expressions

Multiplying rational expressions involves multiplying the numerators and denominators separately. Rational expressions are fractions that involve polynomials. To multiply these fractions:

• Factor all numerators and denominators completely.
• Set up the expression with all factors visible.
• Cancel any common factors from the numerators and denominators.
• Multiply the remaining factors across the numerators and denominators respectively.

Consider the expression given in the problem. After factoring, it becomes \[\frac{(3x + 10)(2x - 5)}{(5x + 2)(3x - 10)} \cdot \frac{(4x + 3)(5x - 2)}{(2x + 5)(x + 2)}\]. With no common factors found to cancel, the expression remains simply connected by multiplication process and reconstruct new rational expression in a factored form.
By multiplying everything as is, you confirm the structure and get a deeper understanding of how rational functions behave.

Simplifying Fractions

Simplifying fractions in algebra involves reducing expressions to their simplest form. This is akin to reducing a fraction like \(\frac{4}{8}\) to \(\frac{1}{2}\). When dealing with rational expressions, follow these steps:

• Factor all components of the expressions thoroughly.
• Cancel any common terms in the numerator and denominator. Only same terms, which means they are identical in form, can be canceled.
• After cancellation, multiply remaining factors to obtain the reduced fraction.

For the given exercise, there were no common factors after factoring, meaning there was no simplification possible during multiplication. The expression after multiplying remains in a factored form as \[\frac{(3x + 10)(4x + 3)(2x - 5)(5x - 2)}{(5x + 2)(3x - 10)(2x + 5)(x + 2)}\]. Here, since the expression is already simplified, you just multiply the factors to display the final expression in its most basic form. Understanding how to simplify such expressions gives you an edge when handling complex algebraic operations.