Problem 15 - Elementary Algebra

Question

Find the LCD. $$ \frac{3}{4 y^{2}-8 y}, \frac{8}{y^{2}-4 y+4}, \frac{10 y-1}{3 y^{3}-6 y^{2}} $$

Step-by-Step Solution

Answer

Answer: The Least Common Denominator of the given fractions is $$4y^3(y - 2)^2$$.

Step 1: Factor the denominators

First, let's factor the denominators: - \(4y^2 - 8y = 4y(y-2)\) - \(y^2 - 4y + 4 = (y - 2)^2\) - \(3y^3 - 6y^2 = 3y^2(y - 2)\)

Step 2: Identify the common factors needed for LCD

Now, we'll determine the common factors needed for the LCD by finding the highest power for each factor present in the denominators. We have: - 4: Appears as a factor in the first term, so the highest power is 1 - \(y\): Appears in all three terms, with the highest power being 3 - \((y-2)\): Appears in all three terms, with the highest power being 2

Step 3: Determine the LCD

Lastly, we'll find the LCD by multiplying the common factors with the highest power: LCD = \(4y^3(y - 2)^2\) Therefore, the Least Common Denominator of the given fractions is: $$4y^3(y - 2)^2$$

Key Concepts

Factorization

Factorization is the process of breaking down a composite number or a polynomial into a product of smaller numbers or polynomials that, when multiplied together, give the original number or polynomial. In algebra, factorization is a vital skill, especially when dealing with rational expressions, also known as algebraic fractions.

For example, factoring the polynomial denominator 4y^2 - 8y, involves finding two polynomials that multiply together to give the original quadratic polynomial. By factoring out the greatest common factor, which is 4y in this case, the expression simplifies to 4y(y - 2). This process reveals the underlying structure of the polynomial, making it easier to work with, especially when determining the least common denominator (LCD) in a set of fractions.

Polynomial Denominators

Polynomial denominators appear in algebraic fractions and represent a more complex form of denominators compared to single terms. They can be monomials, binomials, or trinomials, and more generally, any polynomial expression. To find the LCD for fractions with polynomial denominators, one must first factorize each denominator completely.

In our exercise, after factorization, the denominators become 4y(y - 2), (y - 2)^2, and 3y^2(y - 2). Here, it's important to understand that the LCD for these fractions must contain all the factors of each denominator raised to the highest power that appears in any of the denominators. In educational practices, this ensures that fractions can be added, subtracted, or compared without altering their values.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Operating with these fractions often requires finding the LCD, which allows for the combining of fractions through addition or subtraction.

The LCD is the least common multiple of the denominators. After factoring polynomial denominators, as with the exercise \(\frac{3}{4 y^{2}-8 y}\), \(\frac{8}{y^{2}-4 y+4}\), and \(\frac{10 y-1}{3 y^{3}-6 y^{2}}\), the LCD is determined as 4y^3(y - 2)^2. All fractions can then be adjusted in such a way that each has this LCD as its denominator, simplifying the process of combining the fractions. Understanding algebraic fractions is crucial in advancing algebra skills and tackling a variety of problems in mathematics.