Question

what causes a solution to a rational equation to be an extraneous solution?
when a solution is a fraction, the solution is extraneous.
if a solution results in zero when substituted into the denominator of the equation, the solution is extraneous.
when there is more than one solution, one of the solutions is extraneous.
if a solution results in a negative number when substituted into the denominator of the equation, the solution is extraneous.

Explanation:

An extraneous solution in a rational equation is caused by a solution that makes the denominator of the rational expression equal to zero (since division by zero is undefined). Let's analyze each option:

• The first option is incorrect because a solution being a fraction doesn't make it extraneous.
• The second option is correct: If a solution results in zero when substituted into the denominator of the equation, the solution is extraneous (as it makes the original rational expression undefined).
• The third option is incorrect: Having multiple solutions doesn't inherently make one extraneous; extraneous solutions come from violating the domain (denominator zero), not the number of solutions.
• The fourth option is incorrect: A negative number in the denominator (as long as it's not zero) is allowed, so this doesn't make a solution extraneous.

Answer:

The option: "If a solution results in zero when substituted into the denominator of the equation, the solution is extraneous."