Problem 1 When does a fraction or a ration... [FREE SOLUTION]

Chapter 8: Problem 1

When does a fraction or a rational expression equal \(0 ?\)

Short Answer

Expert verified

A fraction or a rational expression equals 0 when the numerator, P(x), is equal to 0 while the denominator, Q(x), is nonzero for the same x-values. To find these x-values, first solve P(x) = 0 and then ensure that Q(x) is nonzero at those x-values.

Step by step solution

Understand when a fraction is equal to 0

A fraction is equal to 0 if and only if the numerator is equal to 0. The denominator must be nonzero since dividing by 0 is undefined.

Express the fraction as a rational expression

Let's express the fraction as a rational expression using alphabetic notations. Let the numerator be a function P(x) and the denominator be a function Q(x). A rational expression can be written as: \[\frac{P(x)}{Q(x)}\] Now, we will study under which conditions this rational expression equals to 0.

Determine the conditions for the rational expression to be equal to 0

To find when the fraction is equal to 0, we must look for the values of x such that \(\frac{P(x)}{Q(x)} = 0\), given that \(Q(x) \neq 0\). For \(\frac{P(x)}{Q(x)}\) to be equal to 0, the numerator P(x) must be equal to 0. So, for the rational expression to equal 0, we need to find the x-values at which P(x)= 0 while ensuring that Q(x) is not equal to 0 for those x-values. For example, consider the function \[\frac{x^2 - 4}{x - 1}\] We have P(x) = x^2 - 4 and Q(x) = x - 1.

Find the x-values for which P(x) = 0

First, we will find the values for x when P(x) = 0. To do this, we can factor P(x). In our example: P(x) = x^2 - 4 = (x + 2)(x - 2) We can then find the values of x that make the expression true: P(x) = 0 => (x + 2)(x - 2) = 0 So, either x + 2 = 0 or x - 2 = 0, which implies x = -2 or x = 2.

Ensure that Q(x) is nonzero

Now, we need to ensure that at these x-values, Q(x) is not equal to 0. Let's check the behavior of Q(x) for our example: Q(-2) = -2 - 1 = -3 (nonzero) Q(2) = 2 - 1 = 1 (nonzero) So, in our example, the rational expression is equal to 0 when x = -2 or x = 2. In summary, a fraction or a rational expression is equal to 0 when the numerator is equal to 0, and the denominator is nonzero for the same x-values.

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91影视

When does a fraction or a rational expression equal \(0 ?\)

Short Answer

Step by step solution

Understand when a fraction is equal to 0

Express the fraction as a rational expression

Determine the conditions for the rational expression to be equal to 0

Find the x-values for which P(x) = 0

Ensure that Q(x) is nonzero

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