Problem 50 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{2 x^{2}+3 x}{2 x^{3}-10 x^{2}} \cdot \frac{x^{2}-8 x+15}{3 x^{3}-27 x} \div \frac{14 x+21}{x^{2}-6 x-27}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x(x-5)(x-9)}{42x^3(x-1)} \).

Step 1: Factorize the Expressions

Start by factorizing each polynomial in the rational expressions. The first expression is \( \frac{2x^2 + 3x}{2x^3 - 10x^2} \). Factor out common terms: \( 2x^2 + 3x = x(2x + 3) \) and \( 2x^3 - 10x^2 = 2x^2(x - 5) \). Next, for \( \frac{x^2 - 8x + 15}{3x^3 - 27x} \), factor the quadratic and cubic expressions: \( x^2 - 8x + 15 = (x - 3)(x - 5) \) and \( 3x^3 - 27x = 3x(x^2 - 9) = 3x(x - 3)(x + 3) \).Finally, for \( \frac{14x + 21}{x^2 - 6x - 27} \), factor the expressions: \( 14x + 21 = 7(2x + 3) \) and \( x^2 - 6x - 27 = (x - 9)(x + 3) \).

Step 2: Simplify and Convert Division to Multiplication

Replace the division with multiplication by the reciprocal:\[\frac{x(2x + 3)}{2x^2(x - 5)} \cdot \frac{(x - 3)(x - 5)}{3x(x - 3)(x + 3)} \cdot \frac{(x - 9)(x + 3)}{7(2x + 3)}\]

Step 3: Cancel Common Factors

Before multiplying, cancel the common factors across the numerator and denominator:- \(2x + 3\) cancels with \(2x + 3\)- \(x - 3\) cancels with \(x - 3\)- \(x + 3\) cancels with \(x + 3\)This simplifies the expression to:\[\frac{x(x - 5)(x - 9)}{2x^3(3x - 3)(7)}\]

Step 4: Simplify the Final Expression

The expression simplifies to:\[\frac{x(x-5)(x-9)}{42x^3(x-1)}\]Simplify further if possible, but all variables and coefficients are in simplest form.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill when working with rational expressions. It involves breaking down a polynomial into simpler, often linear components, called factors. This can simplify expressions drastically and allow us to see common components that can be canceled out later.
So, how do we factor a polynomial? Think of it similar to splitting a number into its prime factors. For expressions, you'll typically be pulling out a common factor or other easily identifiable terms.
For example, consider the polynomial expression

• \(2x^2 + 3x = x(2x + 3)\), where \(x\) is a common factor.
• For a higher degree polynomial like \(3x^3 - 27x\), factor it as \(3x(x^2 - 9) = 3x(x - 3)(x + 3)\).

By applying this method, you can turn a complicated expression into a collection of simpler, usable terms. This makes manipulation and simplification in further steps much easier.

Simplifying Expressions

Simplifying expressions is all about making expressions as concise and manageable as possible. Ideally, you'd cancel out any terms that appear in both the numerator and denominator.
In our task, we've transformed the operation into:

• \( \frac{x(2x + 3)}{2x^2(x - 5)} \cdot \frac{(x - 3)(x - 5)}{3x(x - 3)(x + 3)} \cdot \frac{(x - 9)(x + 3)}{7(2x + 3)} \).

Now, look for terms that appear in both the numerators and denominators across the entire expression.
Perform the cancellations step-by-step:

• Remove \(2x + 3\) from both sides.
• Cancel out \(x - 3\).
• Eliminate \(x + 3\).

Ultimately, after these cancellations, we end up with a simplified version of the original expression. This approach condenses it, reducing complexity and making it easier to evaluate or use in further calculations.

Operations with Rational Expressions

Handling operations with rational expressions can unfold several layers of arithmetic, primarily multiplication and division. When multiplying, you multiply the numerators together and the denominators together. When dividing, multiply by the reciprocal of the divisor.
For instance:

• Replace any division in the expression with multiplication by flipping the divisor, e.g., \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} \).

This step transforms the problem into manageable chunks, allowing easy manipulation and simplification.
With the expression (before simplification),

• \( \frac{x(x - 5)(x - 9)}{2x^3(3x - 3)(7)} \).

Handle rational expressions meticulously; each algebraic operation is akin to dancing within a choreographed set of moves, ensuring each step follows logically from the last, guiding you smoothly to the simplest form. This ends our expression in its simplest possible shape, ready for interpretation or integration into larger mathematical problems.