Problem 74 - Intermediate Algebra

Question

Perform the operations and simplify. $$ \frac{x^{2}+2 x-35}{12 x^{3}} \cdot \frac{x^{2}+4 x-21}{x^{2}-3 x} $$

Step-by-Step Solution

Answer

Simplified expression is \(\frac{(x+7)(x-5)}{12x^4}\).

Step 1: Factor the Numerators

To simplify the expression, we first factor the numerators of both fractions. For the first fraction, factor the quadratic expression \(x^2 + 2x - 35\). This can be factored into \((x + 7)(x - 5)\). For the second fraction, factor \(x^2 + 4x - 21\) into \((x + 7)(x - 3)\).

Step 2: Factor the Denominators

Next, we factor the denominators of both fractions if possible. The first denominator is simply \(12x^3\), which is in its simplest form. The second denominator is \(x^2 - 3x\), which can be factored as \(x(x - 3)\).

Step 3: Combine the Fractions

Now, we multiply the fractions together: \[ \frac{(x+7)(x-5)}{12x^3} \cdot \frac{(x+7)(x-3)}{x(x-3)} \]. To combine, multiply the numerators and denominators together: \[ \frac{(x+7)(x-5)(x+7)(x-3)}{12x^3 \cdot x(x-3)} \].

Step 4: Simplify the Expression

Before simplifying, cancel any common factors in the numerator and denominator. Notice that \((x+7)(x-3)\) appears in both the numerator and denominator, canceling out directly. The expression simplifies to: \[ \frac{(x+7)(x-5)}{12x^4} \].

Step 5: Final Simplification

The simplified expression \(\frac{(x+7)(x-5)}{12x^4}\) is already in the simplest form since there are no more common factors to cancel.

Key Concepts

Factoring Quadratics

Quadratic expressions are of the form \(ax^2 + bx + c\). Factoring them is a crucial skill to master, especially when dealing with algebraic fractions.
To factor a quadratic expression, you need to find two numbers that multiply to \(c\) (the constant term) and add to \(b\) (the coefficient of \(x\)).
For example, for the expression \(x^2 + 2x - 35\), you want to find two numbers that multiply to \(-35\) and sum to \(2\). These numbers are \(7\) and \(-5\).
Hence, the factored form of \(x^2 + 2x - 35\) is \((x+7)(x-5)\).

• Similarly, for \(x^2 + 4x - 21\), find numbers that multiply to \(-21\) and sum to \(4\). These are \(7\) and \(-3\), making the factors \((x+7)(x-3)\).

Factoring quadratics transforms complex expressions into simpler ones, helping us see patterns and cancel factors conveniently.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials.
The challenge in working with them is performing operations like multiplication and division, followed by simplification.
In our exercise, we're looking at the rational expression \(\frac{x^2+2x-35}{12x^3} \cdot \frac{x^2+4x-21}{x^2-3x}\), which involves both multiplication and simplification.
To multiply rational expressions, you multiply the numerators together and the denominators together. This can lead to complex polynomials, but factoring and canceling can simplify them.

• Numerators: \((x+7)(x-5)\) and \((x+7)(x-3)\).
• Denominators: \(12x^3\) and \(x(x-3)\).

Once the expression is set, look for common factors that can be canceled to make the expression simpler.

Simplifying Expressions

Simplifying expressions is all about making them easier to work with by removing unnecessary complexity. This typically involves canceling common factors and combining like terms.
In our example, after multiplying the expressions, we had: \(\frac{(x+7)(x-5)(x+7)(x-3)}{12x^3 \cdot x(x-3)}\).
Notice how \((x+7)\) and \((x-3)\) appear in both the numerator and denominator.
This commonality allows us to cancel them out, leaving us with \(\frac{(x+7)(x-5)}{12x^4}\).

• Identifying and canceling common factors is the key step in simplifying rational expressions.
• Make sure all expressions are factored completely to ensure no factors are overlooked.

This final step ensures that the expression is in its simplest form, reducing not just its complexity, but also potential errors in further calculations.