Problem 64 - Elementary and Intermediate Algebra

Question

Perform the operations and simplify, if possible. See Example 6 $$\frac{8 y^{2}-14 y-15}{6 y^{2}-11 y-10} \div \frac{4 y^{2}-9 y-9}{3 y^{2}-7 y-6}$$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Rewrite the Division as Multiplication

To divide fractions, we multiply by the reciprocal of the second fraction. Rewrite the expression as: \[\left(\frac{8y^2 - 14y - 15}{6y^2 - 11y - 10}\right) \times \left(\frac{3y^2 - 7y - 6}{4y^2 - 9y - 9}\right)\]

Step 2: Factor All Quadratic Numerators and Denominators

Factor the quadratic expressions. Start with the numerators and denominators:1. \(8y^2 - 14y - 15\) factors to \((4y + 3)(2y - 5)\).2. \(6y^2 - 11y - 10\) factors to \((3y + 2)(2y - 5)\).3. \(3y^2 - 7y - 6\) factors to \((3y + 2)(y - 3)\).4. \(4y^2 - 9y - 9\) factors to \((4y + 3)(y - 3)\).The expression becomes:\[\frac{(4y + 3)(2y - 5)}{(3y + 2)(2y - 5)} \times \frac{(3y + 2)(y - 3)}{(4y + 3)(y - 3)}\]

Step 3: Cancel Common Factors

Identify and cancel the common factors in the numerator and denominator:- \((4y + 3)\) is a common factor.- \((2y - 5)\) is a common factor.- \((3y + 2)\) is a common factor.- \((y - 3)\) is a common factor.After cancelling, no factors are left on the top or the bottom, therefore the expression simplifies to 1.

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial skill in algebra that allows you to break down quadratic expressions into simpler parts. Quadratics are mathematical expressions of the form \(ax^2 + bx + c\). By factoring, we rewrite the expression as a product of two binomials, making it easier to handle. Consider the expression \(8y^2 - 14y - 15\). To factor it effectively, we look for two numbers that multiply to \(a \times c\) (which is \(-120\)) and add up to \(b\) (which is \(-14\)). After identifying these numbers, we break down the middle term and group terms to factor by grouping.
Here’s a short checklist for factoring quadratics:

• Check if there is a common factor in all terms and factor it out first.
• Multiply \(a\) and \(c\), then find two numbers that multiply to this result and add to \(b\).
• Rewrite the middle term using these two numbers and use factoring by grouping to finish.
• Check your factors by redistributing to see if you return to the original quadratic expression.

This step is essential because it transforms complex expressions into simpler ones, easing the process of further operations like division of fractions.

Division of Fractions

Dividing fractions might seem confusing at first, but it can be simplified into a more familiar operation - multiplication. To divide one fraction by another, we multiply by the reciprocal of the divisor. This means we flip the numerator and the denominator of the second fraction. In our example, \[ \frac{8y^2 - 14y - 15}{6y^2 - 11y - 10} \div \frac{4y^2 - 9y - 9}{3y^2 - 7y - 6} \]This is rewritten as:\[ \left(\frac{8y^2 - 14y - 15}{6y^2 - 11y - 10}\right) \times \left(\frac{3y^2 - 7y - 6}{4y^2 - 9y - 9}\right) \]Multiplying by the reciprocal ensures that we're essentially doing the opposite of division. After flipping the second fraction, we proceed to multiply the numerators and denominators. Remember these key steps to successfully divide fractions:

• Flip the second fraction (reciprocal).
• Multiply the numerators together and the denominators together.
• Simplify the new fraction if possible.

This approach simplifies division of fractions to a straightforward multiplication problem. Once you grasp this concept, dividing fractions becomes much more manageable.

Simplifying Expressions

Simplifying expressions is the final step after performing operations like factoring and dividing fractions. It involves reducing expressions to their simplest form. This is achieved by canceling out common factors in the numerators and denominators.
For our example, after factoring the quadratics and rewriting the division as multiplication, we have:\[\frac{(4y + 3)(2y - 5)}{(3y + 2)(2y - 5)} \times \frac{(3y + 2)(y - 3)}{(4y + 3)(y - 3)}\]By examining the expression, identify pairs of factors that appear in both the numerator and the denominator:

• \((4y + 3)\) appears in both top and bottom.
• \((2y - 5)\) appears in both top and bottom.
• \((3y + 2)\) appears in both top and bottom.
• \((y - 3)\) appears in both top and bottom.

Cancel these common factors. If all parts cancel, as in our example, the expression simplifies to 1. Simplifying ensures that you are left with the most reduced, easy to understand form of the expression, making it clean, concise, and easier to interpret.