Problem 56 - Elementary and Intermediate Algebra

Question

Find the LCD of pair of rational expressions. \(\frac{7-y^{2}}{y^{2}-4}, \frac{y-49}{y+2}\)

Step-by-Step Solution

Answer

The LCD is \((y - 2)(y + 2)\).

Step 1: Identify Denominators

Identify the denominators of each rational expression. For the first expression, the denominator is \(y^2 - 4\). For the second expression, the denominator is \(y + 2\).

Step 2: Factor the Denominators

Factor the denominators if possible. The first denominator \(y^2 - 4\) can be factored using the difference of squares: \(y^2 - 4 = (y - 2)(y + 2)\). The second denominator \(y + 2\) cannot be factored further.

Step 3: List All Factors

List all unique factors from both denominators. From \(y^2 - 4\) factored as \((y-2)(y+2)\), we have the factors \(y - 2\) and \(y + 2\). From the second denominator \(y + 2\), we only have \(y + 2\) as a factor.

Step 4: Determine LCD

The least common denominator (LCD) must include each unique factor at its greatest power. The factors are \(y - 2\) and \(y + 2\). Each appears to the power of 1 in this expression. Therefore, the LCD is the product \((y - 2)(y + 2)\).

Key Concepts

Rational Expressions

Rational expressions are like fractions, but instead of integers in the numerator and the denominator, they have polynomials. These expressions can often look quite complex, but remember that as with fractions, the process of simplification and operations like finding a Least Common Denominator (LCD) apply here, too.
To work with rational expressions efficiently, it's crucial to understand some basic terms:

• **Numerator**: The top part of the fraction. In a rational expression, this is a polynomial.
• **Denominator**: The bottom part. Also a polynomial, it determines what values are permissible for the variable used in the expression without making it undefined (denominators cannot be zero).

Mastering operations with rational expressions often involves factoring polynomials, determining the LCD for addition or subtraction, and simplifying.
Understanding these basics makes it easier to handle more complex algebraic problems and prepares you for advanced topics in algebra and calculus.

Factoring Quadratics

Quadratic expressions follow a simple yet fundamental form: \[ ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants. The process of breaking down this quadratic into two simpler binomial expressions is known as factoring. Factoring is critical because it simplifies complex expressions and makes it easier to identify roots.
A common method to factor quadratics is:

• **Search for two numbers that multiply to \(ac\)** and add up to \(b\).
• **Rewrite the middle term** using these two numbers to split the quadratic into four terms.
• **Factor by grouping**: This involves grouping terms in pairs and factoring out common factors within each pair.

However, not all quadratics can be easily factored, especially when they're not perfect squares or don't have rational roots. In those cases, complete the square or use the quadratic formula for solutions. Remember, practice solidifies understanding, so engage with different quadratic equations to master these principles.

Difference of Squares

Difference of Squares is a unique method of factoring applied when you have a specific type of quadratic expression of the form \(a^2 - b^2\). This is because it fits the pattern:\[a^2 - b^2 = (a - b)(a + b)\]This factoring method is particularly neat as it allows for quick simplification. Recognizing the difference of squares is key:

• **Identify perfect squares**: Ensure both terms are perfect squares.
• **Apply the formula**: Substitute into the pattern, which splits the expression into two binomials.

An example from the original exercise is the expression \(y^2 - 4\), which is a difference of squares and factors into \((y - 2)(y + 2)\).
Being able to identify and use the pattern quickly is beneficial for simplifying rational expressions and determining LCDs effectively, as shown in the provided solution. Keep an eye out for similar patterns while solving algebraic expressions.