Algebraic fraction [tex]$\frac{4}{x^2+3 x+2} \div \frac{2}{x^2-1}$[/tex] is given.
Factorize: [tex]$x^2+3 x+2$[/tex] and [tex]$x^2-1$[/tex]
Convert the fraction into the lowest term.
Answer (1)
Factorize the quadratic expressions: x 2 + 3 x + 2 = ( x + 1 ) ( x + 2 ) and x 2 − 1 = ( x + 1 ) ( x − 1 ) .
Rewrite the division as multiplication by the reciprocal: ( x + 1 ) ( x + 2 ) 4 × 2 ( x + 1 ) ( x − 1 ) .
Cancel out the common factors.
The simplified expression is x + 2 2 ( x − 1 ) .
Explanation
Problem Analysis We are given the algebraic fraction x 2 + 3 x + 2 4 ÷ x 2 − 1 2 and asked to simplify it. First, we need to factorize the quadratic expressions in the denominators.
Factorizing the first quadratic expression Let's factorize x 2 + 3 x + 2 . We are looking for two numbers that multiply to 2 and add to 3. These numbers are 1 and 2. Therefore, x 2 + 3 x + 2 = ( x + 1 ) ( x + 2 ) .
Factorizing the second quadratic expression Now, let's factorize x 2 − 1 . This is a difference of squares, so x 2 − 1 = ( x + 1 ) ( x − 1 ) .
Rewriting the division as multiplication Now we can rewrite the original expression using the factorized forms: ( x + 1 ) ( x + 2 ) 4 ÷ ( x + 1 ) ( x − 1 ) 2 To divide fractions, we multiply by the reciprocal of the second fraction: ( x + 1 ) ( x + 2 ) 4 × 2 ( x + 1 ) ( x − 1 )
Canceling common factors Now we can cancel out common factors. We can cancel a factor of ( x + 1 ) from the numerator and denominator, and we can simplify 2 4 to 2: ( x + 2 ) 2 ( x − 1 )
Final simplified expression Therefore, the simplified expression is x + 2 2 ( x − 1 ) .
Examples
Simplifying algebraic fractions is a fundamental skill in algebra and is used in various real-world applications. For example, when designing electrical circuits, engineers often need to simplify complex expressions involving impedances, which can be represented as algebraic fractions. By simplifying these fractions, they can analyze the circuit's behavior more easily and optimize its performance. Similarly, in physics, simplifying algebraic fractions can help in solving problems related to motion, energy, and other physical quantities.
