Enter the correct answer in the box. Write the factored form of the least common denominator needed to simplify this expression.
$\frac{g+1}{g^2+2 g-15}+\frac{g+3}{g+5}$
Answer (1)
Factor the first denominator: g 2 + 2 g − 15 = ( g + 5 ) ( g − 3 ) .
The second denominator is g + 5 .
The least common denominator (LCD) is the product of the unique factors: ( g + 5 ) ( g − 3 ) .
The factored form of the LCD is ( g + 5 ) ( g − 3 ) .
Explanation
Factor the first denominator First, we need to factor the denominator g 2 + 2 g − 15 . We are looking for two numbers that multiply to -15 and add to 2. These numbers are 5 and -3. So, we can factor the quadratic as ( g + 5 ) ( g − 3 ) .
Identify the second denominator The second denominator is already in its simplest form: g + 5 .
Determine the LCD To find the least common denominator (LCD), we take the product of the unique factors from both denominators. The factors are ( g + 5 ) and ( g − 3 ) . Therefore, the LCD is ( g + 5 ) ( g − 3 ) .
State the final answer The factored form of the least common denominator is ( g + 5 ) ( g − 3 ) .
Examples
Understanding how to find the least common denominator is crucial when combining fractions, whether they are numerical or algebraic. For instance, if you're trying to determine how much of two different investments make up your total portfolio, and each investment is represented as a fraction of the total, finding the LCD helps you add those fractions together to see the overall picture. Similarly, in cooking, if you're adjusting recipes and need to combine fractional amounts of ingredients, the LCD ensures accurate measurements and a successful dish. This skill is also fundamental in more advanced math, like calculus, where simplifying complex rational functions is essential for solving integrals and derivatives.
