Problem 29 - Algebra A Combined Function

Question

Find each quotient and simplify. See Examples 4 through 7. $$ \frac{x^{2}+7 x+10}{x-1} \div \frac{x^{2}+2 x-15}{x-1} $$

Step-by-Step Solution

Answer

\( \frac{x+2}{x-3} \)

Step 1: Understanding the problem

The exercise requires you to find the quotient of two rational expressions: \( \frac{x^{2}+7x+10}{x-1} \) and \( \frac{x^{2}+2x-15}{x-1} \). To divide these expressions, we will take the first expression and multiply it by the reciprocal of the second expression.

Step 2: Write the expression using multiplication

Rewrite the division as a multiplication by inverting the divisor. This gives us: \( \frac{x^{2}+7x+10}{x-1} \times \frac{x-1}{x^{2}+2x-15} \). With this form, we can now simplify before multiplying.

Step 3: Simplify the expression

Cancel the \( x-1 \) term from both the numerator of the first fraction and the denominator of the second fraction since they are common factors. This simplifies our expression to \( \frac{x^{2}+7x+10}{x^{2}+2x-15} \).

Step 4: Factor the remaining expressions

Factor the quadratics in the numerator and denominator. The numerator \( x^{2}+7x+10 \) factors into \((x+5)(x+2)\). The denominator \( x^{2}+2x-15 \) factors into \((x+5)(x-3)\).

Step 5: Cancel common factors

Now, simplify the expression \( \frac{(x+5)(x+2)}{(x+5)(x-3)} \) by cancelling the common factor \( x+5 \). This simplifies the expression to \( \frac{x+2}{x-3} \).

Step 6: Verify and express the final simplified quotient

The simplified form of the quotient \( \frac{x+2}{x-3} \) is the final answer, as there are no more common factors to cancel.

Key Concepts

Factoring Quadratics

Factoring quadratics is a key step when simplifying rational expressions. It's like finding the building blocks of a quadratic polynomial. Let’s start with what a quadratic is: any expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, is called a quadratic equation.
The goal is to express it as a product of two binomials, such as \((x + m)(x + n)\). To factor quadratics properly:

• Identify numbers that multiply to \( ac \) and add up to \( b \).
• Split the middle term using these numbers.
• Factor by grouping.

For example, take \( x^2 + 7x + 10 \). You'll need numbers that multiply to 10 and add to 7, which are 5 and 2. Thus, the factors are \((x+5)(x+2)\). Practicing this process makes factoring more intuitive.
Quadratics also appear in both numerators and denominators of rational expressions, so it's essential to become proficient at factoring to simplify later steps.

Division of Rational Expressions

Dividing rational expressions might seem tricky at first, but it’s straightforward with the right approach. It’s similar to the division of fractions, where we multiply by the reciprocal of the divisor.
To divide rational expressions like \( \frac{x^2+7x+10}{x-1} \div \frac{x^2+2x-15}{x-1} \):

• First, write the division as a multiplication by flipping the second fraction.
• This transforms the expression to \( \frac{x^2+7x+10}{x-1} \times \frac{x-1}{x^2+2x-15} \).
• Now, you can easily cancel terms or simplify before multiplying.

The concept of reciprocal transformation allows for a more straightforward cancellation of common terms, leveraging the multiplication format to simplify otherwise complex rational expressions. This technique is helpful, especially when simplifying is needed across significant expressions.

Simplification Techniques

Simplification is crucial for making complex mathematical expressions easier to understand and work with. When simplifying rational expressions, the key steps involve factorization and cancellation.
Start by canceling any common factors in the numerator and the denominator before multiplying. For instance:

• In our given expression, cancel \( x-1 \) from the items \( \frac{x^2+7x+10}{x-1} \times \frac{x-1}{x^2+2x-15} \) because it's a common factor.
• This simplifies it to \( \frac{x^2+7x+10}{x^2+2x-15} \).
• Then, factor the quadratic expressions as previously explained.
• Finally, cancel any other common factors such as \( x+5 \) in this case, resulting in a simpler expression \( \frac{x+2}{x-3} \).

Simplification reduces the complexity of expressions, making them not only shorter but easier to work with and understand. Always remember, the goal is to make an expression as simplified as possible without altering its value.