Question

First see if you can cancel factors in the numerator and denominator. Then, using factors from the denominator make an equation stating that the fraction = A/(factor₁) + B/(factor₂) where A and B are constants. Multiply both sides by the denominator ((factor₁)×(factor₂)) to give the numerator = A(factor₂) + B(factor₁), then by substituting x = root₁, we can calculate the value of B (since x−root₁ will now clear the term with A), and by substituting x = root₂, we can calculate the value of A.

What is the first step when simplifying a rational expression with a fraction of the form (numerator)/(denominator)?
A) Cancel common factors in both the numerator and denominator.
B) Find the least common denominator.
C) Expand the numerator and denominator.
D) Take the reciprocal of the fraction.

Answer 1

The question involves both partial fraction decomposition and the more straightforward process of multiplying fractions by multiplying numerators and denominators, followed by simplification. A common denominator is necessary for adding or subtracting fractions. Units of measurement within fractions must be correctly canceled to ensure accurate results.

Step-by-step explanation:

The student's question covers the process of partial fraction decomposition which is typically used when working with algebraic fractions. Essentially, when you have a single fraction that can be split into multiple fractions with simpler denominators, you would want to break it down using constants for easier integration or simplification. However, the initial part of the student's question involves multiplying fractions where the numerators are multiplied together, and the denominators are multiplied together, often simplifying by common factors if possible.

An important step in adding or subtracting fractions is finding a common denominator. After this, the numerators can be easily added or subtracted while the denominator remains the same. This method ensures that the integrity of the fractions is maintained and does not lead to nonsense answers.

For example, multiplying the fraction 1/5 by the number 3 using the straight rules would give us 3 in the numerator and 5 in the denominator. Many common factors might appear during this process, and it is important to simplify as much as possible to arrive at the final answer, which in this case would remain 3/5. Furthermore, when considering units of measurement in fractions, proper cancellation of units will leave only the desired units, making sure the answer's units are correct.

Overall, understanding how to manipulate and simplify fractions is crucial when dealing with calculations in Mathematics, especially at the high school level, and following the correct steps ensures accurate results.