Problem 82 - Intermediate Algebra

Question

Perform the operations and simplify. $$ \frac{2 p^{2}-5 p q-3 q^{2}}{p^{2}-9 q^{2}} \div \frac{2 p^{2}+5 p q+2 q^{2}}{2 p^{2}+5 p q-3 q^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{(2p + 3q)(p - q)}{(p + 3q)(p + 2q)} \).

Step 1: Understand the expression

The given expression is a division of two rational expressions: \( \frac{2 p^{2}-5 p q-3 q^{2}}{p^{2}-9 q^{2}} \div \frac{2 p^{2}+5 p q+2 q^{2}}{2 p^{2}+5 p q-3 q^{2}} \). To simplify, recall that dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 2: Convert division to multiplication

Rewrite the division as multiplication by the reciprocal. Thus, the expression becomes \( \frac{2 p^{2}-5 p q-3 q^{2}}{p^{2}-9 q^{2}} \times \frac{2 p^{2}+5 p q-3 q^{2}}{2 p^{2}+5 p q+2 q^{2}} \).

Step 3: Factor the expressions (1)

Factor the numerators and denominators, if possible. Start with \(2 p^{2} - 5 p q - 3 q^{2}\), which factors as \((2p + q)(p - 3q)\).

Step 4: Factor the expressions (2)

Next, factor \(p^{2} - 9 q^{2}\) using the difference of squares: \((p + 3q)(p - 3q)\).

Step 5: Factor the expressions (3)

For the expression \(2 p^{2}+5 p q+2 q^{2}\), it factors as \((2p + q)(p + 2q)\).

Step 6: Factor the expressions (4)

Finally, factor \(2 p^{2}+5 p q-3 q^{2}\) as \((2p + 3q)(p - q)\).

Step 7: Simplify the expression

Substitute back the factors: \[ \frac{(2p + q)(p - 3q)}{(p + 3q)(p - 3q)} \times \frac{(2p + 3q)(p - q)}{(2p + q)(p + 2q)} \]. Cancel out common factors: \(2p+q\) and \(p - 3q\).

Step 8: Complete the simplification

After cancelation, we have: \( \frac{(2p + 3q)(p - q)}{(p + 3q)(p + 2q)} \). Verify if further simplification can be done; none can be observed.

Key Concepts

Division of Rational Expressions

Dividing rational expressions might seem complex, but it can be simplified by using the concept of reciprocals. To divide one rational expression by another, we simply multiply the first expression by the reciprocal of the second. This is much like the way we handle division of fractions in arithmetic. Here’s how it works:

- **Identify the division**: When you see a division symbol between two fractional expressions, get ready to transform it. - **Flip the divisor**: Switch the numerator and denominator of the expression it's divided by. This turns it into its reciprocal. - **Multiply**: Change the division to multiplication and multiply the resulting expressions. By following these steps, the initial division turns into a multiplication problem, which is much easier to handle and simplifies the process. Always remember, turning a division problem into a multiplication problem can help in managing complex expressions.

Factoring Polynomials

Factoring polynomials is essential when working with rational expressions because it allows us to break down complex expressions into simpler multiplicative components. This process makes it easier to cancel out terms and simplify the expression.

Here are the main approaches in factoring:

• **Factoring out common terms**: Look for common factors in each term of the polynomial and factor them out.
• **Difference of squares**: Recognize expressions in the form of \(a^2 - b^2\) and factor them as \((a - b)(a + b)\).
• **Quadratic trinomials**: For expressions of the form \(ax^2 + bx + c\), find two numbers that multiply to \(ac\) and add to \(b\), then use these numbers to split the middle term and factor by grouping.

In our example, we factored several polynomials using these techniques:- \(2p^2 - 5pq - 3q^2\) is factored into \((2p + q)(p - 3q)\).- \(p^2 - 9q^2\) is a difference of squares, factored as \((p + 3q)(p - 3q)\).By understanding and applying these techniques, you can simplify complex rational expressions efficiently.

Simplifying Rational Expressions

The final step in working with rational expressions is the simplification process, which involves canceling out common factors. When the numerators and denominators of rational expressions are factored, simplifying becomes straightforward.

This is how you can simplify rational expressions:

• **Cancel common factors**: Once factored, identify and remove any common factors that appear in both the numerator and the denominator.
• **Re-evaluate**: After cancelation, assess the remaining expression to see if any further simplifications are possible.

In our problem, we recognized common factors after factoring:- The factors \(2p + q\) and \(p - 3q\) appeared in both the numerator and denominator. This allowed us to cancel them, leading to the simplified expression: \(\frac{(2p + 3q)(p - q)}{(p + 3q)(p + 2q)}\).Simplifying rational expressions can greatly reduce the complexity of the problem, allowing for clearer insights into its nature. With practice, identifying and canceling out these factors become second nature, making these expressions much easier to handle.