Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, divide the rational expressions. \(\frac{16 a^{2}-24 a+9}{4 a^{2}+17 a-15} \div \frac{16 a^{2}-9}{4 a^{2}+11 a+6}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4a^2 + 5a - 6}{a + 5} \).

Step 1: Identify the Division Problem

We are given the expression \( \frac{16 a^{2}-24 a+9}{4 a^{2}+17 a-15} \div \frac{16 a^{2}-9}{4 a^{2}+11 a+6} \). Since dividing by a fraction is the same as multiplying by its reciprocal, we'll convert the division into a multiplication problem.

Step 2: Convert Division to Multiplication

Rewrite the expression by multiplying the first rational expression by the reciprocal of the second: \( \frac{16 a^{2}-24 a+9}{4 a^{2}+17 a-15} \times \frac{4 a^{2}+11 a+6}{16 a^{2}-9} \).

Step 3: Factor All Expressions

Factor each numerator and denominator: - \(16a^2 - 24a + 9\) factors to \((4a - 3)^2\). - \(4a^2 + 17a - 15\) factors to \((4a - 3)(a + 5)\). - \(16a^2 - 9\) is a difference of squares and factors to \((4a - 3)(4a + 3)\). - \(4a^2 + 11a + 6\) factors to \((4a + 3)(a + 2)\).

Step 4: Substitute Factored Forms into the Expression

Substitute the factored forms into the multiplication: \( \frac{(4a - 3)^2}{(4a - 3)(a + 5)} \times \frac{(4a + 3)(a + 2)}{(4a - 3)(4a + 3)} \).

Step 5: Simplify the Expression

Cancel out the common factors in the numerator and the denominator. We can cancel one \(4a - 3\) from the numerator and denominator and \(4a + 3\) from the multiplication: - Remaining expression: \( \frac{4a - 3}{a + 5} \times \frac{a + 2}{1} \).

Step 6: Multiply the Remaining Expressions

Combine the remaining factors to get: \( \frac{(4a - 3)(a + 2)}{a + 5} \).

Step 7: Expand If Required

Expand if necessary. Multiply \((4a - 3)(a + 2)\) to expand it further to \(4a^2 + 8a - 3a - 6 = 4a^2 + 5a - 6\). The expression becomes \( \frac{4a^2 + 5a - 6}{a + 5} \).

Key Concepts

Division of Rational Expressions

When dividing rational expressions, the process is quite similar to the division of regular fractions. You can make it easier by converting the division operation into multiplication. To do this, you take the reciprocal of the expression you are dividing by. For example, if you need to divide one rational expression by another, such as \( \frac{A}{B} \div \frac{C}{D} \), you can rewrite it as \( \frac{A}{B} \times \frac{D}{C} \). This change allows you to tackle the problem using multiplication rules, simplifying the process significantly. Just remember to flip the second fraction and multiply, which brings us to the next step—factoring.

Factoring Polynomials

Factoring is a technique used to simplify polynomial expressions into their more basic components. It involves expressing a polynomial as a product of its factors. In our problem, each polynomial needs to be factored correctly before further simplification. For instance:

• The trinomial \(16a^2 - 24a + 9\) can be factored into \((4a - 3)^2\).
• The trinomial \(4a^2 + 17a - 15\) factors into \((4a - 3)(a + 5)\).

If you encounter a difference of squares, such as \(16a^2 - 9\), apply the formula \(a^2 - b^2 = (a - b)(a + b)\) to get \((4a - 3)(4a + 3)\). Factoring is crucial as it helps identify and cancel common factors, which simplifies the expression further.

Multiplication of Rational Expressions

After factoring, the next step is to multiply rational expressions. Multiplication is straightforward—multiply the numerators together and then multiply the denominators together. Here, we multiplied:

• The factored form of the first rational expression \(\frac{(4a - 3)^2}{(4a - 3)(a + 5)}\)
• By the reciprocal of the second \(\frac{(4a + 3)(a + 2)}{(4a - 3)(4a + 3)}\)

This results in a single rational expression, but not yet simplified. Use multiplication to combine elements, then simplify where possible by canceling out common terms.

Simplifying Rational Expressions

Simplification is the final objective in handling rational expressions. It involves canceling out common factors from the numerator and denominator. During simplification, look for repeating factors and cancel them out, which was done in our problem by canceling out \((4a - 3)\) and \((4a + 3)\) from both the numerator and denominator. You're left with the expression:

• \(\frac{4a - 3}{a + 5} \times \frac{a + 2}{1}\)

Finally, by multiplying the remaining terms, you get \(\frac{(4a - 3)(a + 2)}{a + 5}\). This can be expanded if necessary for a final expanded form of \(\frac{4a^2 + 5a - 6}{a + 5}\). Simplifying helps in managing and solving rational expressions effectively, providing a more concise and usable form.