Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, divide the rational expressions. \(\frac{144 b^{2}-25}{72 b^{2}-6 b-10} \div \frac{18 b^{2}-21 b+5}{36 b^{2}-18 b-10}\)

Step-by-Step Solution

Answer

The simplified result is \( \frac{(12b - 5)(12b + 5)(6b + 1)}{(4b - 5)(9b + 1)(3b - 1)} \).

Step 1: Identify the Operation

The operation is division of two rational expressions: \( \frac{144 b^{2}-25}{72 b^{2}-6 b-10} \div \frac{18 b^{2}-21 b+5}{36 b^{2}-18b-10} \). To divide fractions, we multiply by the reciprocal of the divisor.

Step 2: Write the Reciprocal

The reciprocal of the second expression \( \frac{18 b^{2}-21 b+5}{36 b^{2}-18b-10} \) is \( \frac{36 b^{2}-18b-10}{18 b^{2}-21 b+5} \). Thus, the expression becomes \( \frac{144 b^{2}-25}{72 b^{2}-6b-10} \times \frac{36 b^{2}-18b-10}{18 b^{2}-21 b+5} \).

Step 3: Factor Each Quadratic Expression

Factor each of the quadratic expressions:- \(144 b^2 - 25\) is a difference of squares: \((12b - 5)(12b + 5)\).- \(72 b^2 - 6b - 10\) simplifies to \(2(36b^2 - 3b - 5)\) and factors to \((4b - 5)(9b + 1)\).- \(36 b^2 - 18b - 10\) can be factored as \((6b - 5)(6b + 1)\).- \(18 b^2 - 21b + 5\) factors to \((3b - 1)(6b - 5)\).

Step 4: Simplify the Expression by Cancelling Common Factors

Replace each expression with its factors: \[ \frac{(12b - 5)(12b + 5)}{(4b - 5)(9b + 1)} \times \frac{(6b - 5)(6b + 1)}{(3b - 1)(6b - 5)} \].Cancel out the common factors across numerators and denominators: - \((6b - 5)\) in the numerator and denominator.The expression simplifies to:\[ \frac{(12b - 5)(12b + 5)(6b + 1)}{(4b - 5)(9b + 1)(3b - 1)} \].

Key Concepts

Factoring Quadratics

Factoring quadratics is the process of breaking down a quadratic expression into simpler expressions, typically a product of linear factors. Quadratic expressions appear in the form \(ax^2 + bx + c\). Here, factoring involves finding two binomials that multiply to give the original quadratic. This can make operations like dividing or multiplying expressions much simpler. To factor a quadratic:

• Look for common factors. Always start by checking if there is a number or variable that is common to all terms.
• For quadratics of the form \(x^2 + bx + c\), look for two numbers that multiply to \(c\) and add to \(b\).
• For more complex quadratics, such as \(ax^2 + bx + c\) where \(a eq 1\), consider using factoring by grouping or the quadratic formula if necessary.

Difference of Squares

The difference of squares is a special type of quadratic expression in the form \(a^2 - b^2\). It can be easily factored into \((a - b)(a + b)\). This pattern is particularly useful when simplifying expressions or solving equations where the quadratic can be seen as a perfect square subtracted from another.Identifying a difference of squares:

• Ensure the expression is in the form \(a^2 - b^2\).
• Recognize squares: this means identifying terms like \(x^2\), \((3y)^2\), \(25\), etc.
• Once identified, apply the factorization \((a - b)(a + b)\) for streamlined calculations.

Reciprocal of a Fraction

The reciprocal of a fraction is simply flipping the numerator and the denominator. In mathematical operations, when dividing by a fraction, you multiply by its reciprocal. This shifts the operation from division to multiplication, simplifying the process.How to find the reciprocal:

• Take the fraction \(\frac{a}{b}\).
• Flip it to become \(\frac{b}{a}\).
• Use this in division problems by changing the division to multiplication with the reciprocal of the divisor.

This is crucial for dividing rational expressions, allowing the calculations to focus on straightforward multiplication.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. This typically means factoring numbers and variables, canceling out terms, and making the expression more manageable. Steps to simplify rational expressions:

• Factor all numerators and denominators into their simplest parts.
• Identify and cancel out common factors in the numerator and denominator.
• Re-write the expression to reflect these cancellations for the simplest form.

The aim is to present the expression as clearly as possible, which both simplifies understanding and sets the stage for easier further calculations.