Problem 57
Question
For the following problems, perform the multiplications and divisions. $$ \frac{a^{2}-4 a-12}{a^{2}-9} \div \frac{a^{2}-5 a-6}{a^{2}+6 a+9} $$
Step-by-Step Solution
Verified Answer
Question: Given the division problem below, simplify the rational expressions:
\(\frac{a^2 - 4a - 12}{a^2 - 9} \div \frac{a^2 - 5a - 6}{a^2 + 6a + 9}\)
Answer: \(\frac{(a + 2)(a + 3)^2}{(a - 3)(a + 3)(a + 1)}\)
1Step 1: Factor numerators and denominators
Factor the numerators and denominators as follows:
First expression: \(\frac{a^2 - 4a - 12}{a^2 - 9}\)
Numerator: \(a^2 - 4a - 12 = (a - 6)(a + 2)\)
Denominator: \(a^2 - 9 = (a - 3)(a + 3)\)
So the first expression becomes: \(\frac{(a - 6)(a + 2)}{(a - 3)(a + 3)}\)
Second expression: \(\frac{a^2 - 5a - 6}{a^2 + 6a + 9}\)
Numerator: \(a^2 - 5a - 6 = (a - 6)(a + 1)\)
Denominator: \(a^2 + 6a + 9 = (a + 3)^2\)
So the second expression becomes: \(\frac{(a - 6)(a + 1)}{(a + 3)^2}\)
2Step 2: Find the reciprocal of the second expression
Reciprocal of the second expression is: \(\frac{(a + 3)^2}{(a - 6)(a + 1)}\)
3Step 3: Change division operation to multiplication
Now change the division operation to multiplication by taking the reciprocal of the second expression:
$$\frac{(a - 6)(a + 2)}{(a - 3)(a + 3)} \times \frac{(a + 3)^2}{(a - 6)(a + 1)}$$
4Step 4: Cancel common factors
In both expressions, we have a common factor of \((a - 6)\):
$$\frac{(a + 2)}{(a - 3)(a + 3)} \times \frac{(a + 3)^2}{(a + 1)}$$
5Step 5: Multiply numerators and denominators
Now multiply both numerators together and both denominators together:
$$\frac{(a + 2)(a + 3)^2}{(a - 3)(a + 3)(a + 1)}$$
The final result of the expression is:
$$\frac{(a + 2)(a + 3)^2}{(a - 3)(a + 3)(a + 1)}$$
Key Concepts
Factoring QuadraticsRational ExpressionsMultiplication and Division of Fractions
Factoring Quadratics
Factoring quadratics is an essential skill in algebra that involves breaking down a quadratic expression, generally of the form \( ax^2 + bx + c \), into products of simpler binomials. This process is key in simplifying fractions or solving equations. Let's take a closer look at an example to better understand the process.
To factor the quadratic expression \( a^2 - 4a - 12 \), we aim to express it in the form \((a - 6)(a + 2)\). Similarly, for \( a^2 - 9 \), it is factored as \((a - 3)(a + 3)\), known as a difference of squares.
To factor the quadratic expression \( a^2 - 4a - 12 \), we aim to express it in the form \((a - 6)(a + 2)\). Similarly, for \( a^2 - 9 \), it is factored as \((a - 3)(a + 3)\), known as a difference of squares.
- Identify two numbers that multiply to give the product of \( a \cdot c \) (here \( -12 \)) and add to give \( b \) (here \( -4 \)).
- Use these numbers to split the middle term and factor by grouping.
Rational Expressions
Rational expressions are fractions that include polynomials in the numerator and denominator. They resemble regular fractions but need special handling due to the variables. Simplifying them often involves factoring to find common factors that can be canceled.
Let's consider an expression like \( \frac{a^2 - 4a - 12}{a^2 - 9} \). After factoring as explained previously, it simplifies into \( \frac{(a - 6)(a + 2)}{(a - 3)(a + 3)} \). Rational expressions require:
Let's consider an expression like \( \frac{a^2 - 4a - 12}{a^2 - 9} \). After factoring as explained previously, it simplifies into \( \frac{(a - 6)(a + 2)}{(a - 3)(a + 3)} \). Rational expressions require:
- Ensuring the denominator is never zero, as division by zero is undefined.
- Finding the domain of the expression by identifying values that make the denominator zero.
Multiplication and Division of Fractions
Multiplying and dividing fractions is crucial in algebra when dealing with rational expressions. The process may be slightly different from numerical fractions because of the variable components.
To divide fractions, like in the expression given here, we first take the reciprocal of the second fraction and change the division sign to multiplication. For example, \( \frac{(a + 3)^2}{(a - 6)(a + 1)} \) becomes \( \frac{(a - 6)(a + 1)}{(a + 3)^2} \) in reciprocal form. Thus, \( \frac{ (a - 6)(a + 2)}{(a - 3)(a + 3)} \times \frac{(a + 3)^2}{(a - 6)(a + 1)} \) is what we compute.
To divide fractions, like in the expression given here, we first take the reciprocal of the second fraction and change the division sign to multiplication. For example, \( \frac{(a + 3)^2}{(a - 6)(a + 1)} \) becomes \( \frac{(a - 6)(a + 1)}{(a + 3)^2} \) in reciprocal form. Thus, \( \frac{ (a - 6)(a + 2)}{(a - 3)(a + 3)} \times \frac{(a + 3)^2}{(a - 6)(a + 1)} \) is what we compute.
- Cancel out similar terms in the numerator and denominator to simplify.
- Multiply the remaining numerators and denominators individually.
- An important concept is that multiplying by a reciprocal effectively cancels common factors between expressions.
Other exercises in this chapter
Problem 57
For the following problems, perform the divisions. $$ \frac{a^{2}+5 a+4}{a^{2}-a-2} $$
View solution Problem 57
For the following problems, solve the rational equations. $$ \frac{2}{x}=\frac{3}{x+2}+1 $$
View solution Problem 57
For the following problems, add or subtract the rational expressions. $$ \frac{y+2}{(y+1)(y+6)}+\frac{y-2}{y+6} $$
View solution Problem 57
For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{4 x-7}{-7}\)
View solution