Problem 30 - Algebra and Trigonometry

Question

multiply or divide as indicated. $$ \frac{x^{2}-4}{x^{2}+3 x-10} \div \frac{x^{2}+5 x+6}{x^{2}+8 x+15} $$

Step-by-Step Solution

Answer

The quotient of this division is \(x^{2} + 3x - 10\).

Step 1: Factor the Quadratic Expressions

Each quadratic expression within the fractions should be factored into simpler expressions. \n(i) The expression \(x^{2}-4\) can be written as \((x-2)(x+2)\). \n(ii) The expression \(x^{2}+3x-10\) can be written as \((x-5)(x+2)\). \n(iii) The expression \(x^{2}+5x+6\) can be written as \((x+2)(x+3)\). \n(iv) The expression \(x^{2}+8x+15\) can be written as \((x+5)(x+3)\). Thus, with substitution, the expression becomes: \(\frac{(x-2)(x+2)}{(x-5)(x+2)} \div \frac{(x+2)(x+3)}{(x+5)(x+3)}\)

Step 2: Change Division to Multiplication and Multiply the Fractions

It’s easier to change the division operation to multiplication by reciprocating the divisor fraction. The expression becomes: \(\frac{(x-2)(x+2)}{(x-5)(x+2)} \times \frac{(x+5)(x+3)}{(x+2)(x+3)}\). Now, perform the multiplication operation. When multiplying fractions, same terms in the numerator and the denominator would cancel out. Thus, the \(x+2\) and \(x+3\) terms will cancel out: \(\frac{(x-2)\cancel{(x+2)}}{\cancel{(x-2)}\cancel{(x+3)}} \times \frac{(x+5)\cancel{(x+3)}}{\cancel{(x+2)}\cancel{(x+3)}}\). So the expression becomes: \(\frac{x-2}{1} \times \frac{x+5}{1}\). This further reduces to \((x-2)(x+5)\).

Step 3: Expand Expression

Expand the expression \((x-2)(x+5)\) by performing the multiplication. This is done by multiplying each term in the first bracket by each term in the second bracket. Hence, \(x \times x\), \(x \times 5\), \(-2 \times x\), \(-2 \times 5\) which gives \(x^{2} + 5x - 2x - 10\). Combining like terms results in \(x^{2} + 3x - 10\).

Key Concepts

Factoring Quadratic Expressions

Understanding how to factor quadratic expressions is essential before tackling operations with rational expressions. Factoring is the process of breaking down a complicated expression into products of simpler ones. In the context of quadratics, which are expressions of the form \(ax^2 + bx + c\), factoring often involves finding two binomials that multiply together to give the original quadratic.

As an example, consider \(x^2 - 4\), which is a difference of squares and can be factored into \((x - 2)(x + 2)\). This form is much easier to manage, especially when involved in fractions and complex algebraic operations. Looking for patterns like the difference of squares, the sum and product method for trinomials, or even the quadratic formula when necessary, can help you factor these expressions effectively.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Like numeric fractions, these expressions can be simplified, added, subtracted, multiplied, and divided. Key to working with them is the factorization of the polynomials, which allows for the identification of common factors that can be canceled out to simplify the expression.

When multiplying or dividing rational expressions, factoring becomes incredibly important, as seen in the exercise where factored forms allow for the cancellation of common factors, simplifying the complex fraction to much simpler terms. Simplification can reduce a seemingly daunting algebra problem to something much more straightforward.

Simplifying Complex Fractions

Complex fractions may look intimidating, but they share the same principles as simple fractions. To simplify a complex fraction, the first step is often to factor the numerator and the denominator, as shown in the exercise. Then, if you encounter a division sign, you'll turn it into a multiplication by flipping the second fraction, which is known as finding the reciprocal.

The next step is cancellation. If the same factor appears in both a numerator and a denominator, they can be canceled out. Once all possible cancellations are made, you may be left with a much simpler expression, which can often be a single fraction or even a whole number. This technique cuts through the complexity and makes the problem far more manageable.

Algebraic Operations

Algebraic operations such as addition, subtraction, multiplication, and division are fundamental to working with rational expressions. Proficiency with these operations is crucial for performing the next step in the simplification process. Multiplication and division, in particular, require a firm understanding of how to manage polynomials.

As seen in the solution to the given exercise, after changing division to multiplication and factoring, the next step is to multiply across the numerators and denominators, followed by simplifying and canceling where appropriate. When expanding binomials or more complex expressions, it's important to multiply each term in one polynomial by each term in the other, like \((x - 2)(x + 5)\) becoming \(x^2 + 3x - 10\) after simplification. Mastering these operations takes practice but is immensely rewarding when solving algebraic equations and rational expressions.