Are a rational expression and its simplified forms equivalent?
Provide mathematical examples to support your opinions. You may use tables or graphs to organize and present your thoughts.
Can somebody give me some guidance for answering this question or provide me with a good site so that I can understand? Appreciated, thanks
(x+2)(x-3)
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(x+2)
This is equal to (x-3) for every value of x except -2, where it is undefined.
To determine whether a rational expression and its simplified forms are equivalent, we need to understand the concept of simplifying rational expressions and how to compare expressions for equivalence.
A rational expression is a fraction wherein both the numerator and denominator are polynomial expressions. Simplifying a rational expression involves reducing it to its simplest form by canceling out common factors, removing any unnecessary parentheses, and reducing the expression to its lowest terms.
To prove that a rational expression and its simplified forms are equivalent, we need to show that they have the same value for all valid inputs.
Let's consider an example to illustrate this concept:
Example 1:
Consider the rational expression (3x^2 + 6x) / (x^3 + 3x^2).
To simplify it, we can factor out a common term of x from both the numerator and denominator, resulting in:
(3x(x + 2)) / (x^2(x + 3)).
By canceling out the common factor of (x + 2), we obtain the simplified form:
3x / (x^2 + 3x).
To check if the original and simplified forms are equivalent, we can evaluate both expressions for various values of x and compare the results. For example, let's evaluate both expressions for x = 2:
Original form: (3(2)^2 + 6(2)) / ((2)^3 + 3(2)^2) = 18 / 20 = 9 / 10
Simplified form: 3(2) / ((2)^2 + 3(2)) = 6 / 10 = 3 / 5
Since the original and simplified forms give the same result for x = 2, we can conclude that they are equivalent for that particular value of x.
To understand this topic more comprehensively and solve additional examples, you may refer to reputable online math resources such as Khan Academy (www.khanacademy.org) or Mathway (www.mathway.com). These websites provide video lessons, step-by-step guidance, and interactive tools to help you understand and solve problems related to rational expressions.