Problem 28 - College Algebra

Question

For the following exercises, divide the rational expressions. $$ \frac{16 x^{2}+18 x-55}{32 x^{2}-36 x-11} \div \frac{2 x^{2}+17 x+30}{4 x^{2}+25 x+6} $$

Step-by-Step Solution

Answer

\( \frac{(2x - 5)(4x + 1)(8x + 11)}{(4x - 11)(8x + 1)(2x + 5)} \) is the simplified result.

Step 1: Rewrite Division as Multiplication

Convert the division of the two rational expressions into multiplication by flipping the second fraction. Start with: \( \frac{16x^2 + 18x - 55}{32x^2 - 36x - 11} \div \frac{2x^2 + 17x + 30}{4x^2 + 25x + 6} \). This becomes \( \frac{16x^2 + 18x - 55}{32x^2 - 36x - 11} \times \frac{4x^2 + 25x + 6}{2x^2 + 17x + 30} \).

Step 2: Factor All Quadratic Expressions

Factor each quadratic expression involved in the multiplication. - \( 16x^2 + 18x - 55 = (2x - 5)(8x + 11) \)- \( 32x^2 - 36x - 11 = (4x - 11)(8x + 1) \)- \( 2x^2 + 17x + 30 = (2x + 5)(x + 6) \)- \( 4x^2 + 25x + 6 = (4x + 1)(x + 6) \).

Step 3: Set Up the Factored Expression

Substitute the factored forms back into the expression: \( \frac{(2x - 5)(8x + 11)}{(4x - 11)(8x + 1)} \times \frac{(4x + 1)(x + 6)}{(2x + 5)(x + 6)} \).

Step 4: Cancel Common Factors

Identify and cancel out any common factors from the numerators and denominators across the entire expression. - Cancel \((x + 6)\) from both numerator and denominator.- Then the expression reduces to \( \frac{(2x - 5)(8x + 11)(4x + 1)}{(4x - 11)(8x + 1)(2x + 5)} \).

Step 5: Simplify the Expression

The remaining expression after canceling common factors is already simplified: \( \frac{(2x - 5)(4x + 1)(8x + 11)}{(4x - 11)(8x + 1)(2x + 5)} \). There are no further common terms to cancel.

Key Concepts

Factoring Quadratics

When dealing with rational expressions, factoring quadratics is an essential skill. Quadratic equations often appear in the form of \( ax^2 + bx + c \). The goal is to express the quadratic as a product of two binomials. This step simplifies the rational expressions and helps identify common factors that can be canceled later in the process.

To factor quadratics effectively, follow these steps:

• Identify the coefficents \( a \), \( b \), and \( c \) in the quadratic expression \( ax^2 + bx + c \).
• Look for two numbers that multiply to the product \( ac \) and add up to \( b \).
• Rewrite the middle term, \( bx \), using the two numbers found, essentially splitting \( b \) into two parts.
• Factor by grouping the terms and finding common factors within each group.

Factoring might seem challenging at first, but with practice, it becomes an intuitive process. Mastery of factoring will allow you to work fluently with more complex algebraic expressions.

Division of Rational Expressions

Division of rational expressions involves converting the division problem into a multiplication problem. This is achieved by multiplying by the reciprocal of the divisor. In rational expressions, you begin with two fractions and "flip" the second fraction before multiplying.

Here's how you can do it:

• Start with the division of two rational expressions, for example, \( \frac{A}{B} \div \frac{C}{D} \).
• Rewrite the division as \( \frac{A}{B} \times \frac{D}{C} \), changing the operation to multiplication and flipping the second fraction.
• Once rewritten, consider this setup as a single multiplication expression, which can be simplified through factoring and canceling common factors.

In the given example, by shifting from division to multiplication, simplification became more accessible by allowing the factoring and cancellation of terms.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means reducing them to their simplest form. In rational expressions, this process involves canceling out common factors in the numerator and denominator. This not only makes expressions easier to work with but also reveals the simplest ratio of the involved terms.

Follow these steps to simplify:

• Factor both the numerator and denominator completely to identify common factors.
• Cancel those common factors across the numerator and the denominator since they effectively divide into one another.
• Check your final expression to ensure no further simplification is possible.

When you simplify, always ensure you’ve transformed the rational expression accurately, preserving the values within its domain. This practice ensures that you're working with the expression's simplest and most efficient form.