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Hello,

I am a first-year math grad student, and I just started teaching two sections of precalc! yikes!

So I had a major embarrasing moment during office hours when I told 2 students the wrong answer because I completely forgot about the phenomenon known as "Extraneous roots"! Does anyone remember this from grade school?

For example, if I have x^(1/4)=-2, putting both sides to the 4th power will give me x=16, but this is an incorrect solution, as 16^(1/4)=2. (EDIT: This is with the convention that roots are well-defined functions, so that the only give you the positive answer. I think if you see an equation to solve with radicals involved, it is assumed that the radicals mean positive roots). While this example is simple enough, there are cases where it is not obvious that I have an extraneous root until I plug-in to check!

I am trying to create a list of operations that cause equations to gain additional roots (or lose them). So far, I have:

Gain extraneous roots by:

1. Multiplying both sides by an expression containing a variable. (common for solving rational equations).

2. Raising both sides to an even power.

Lose roots by:

1. Dividing both sides by an expression containing a variable.

2. Taking the 2n-th square root on both sides (unless you remember to put the obligatory plus-minus sign in front).

Any others I am missing? I have consulted various algebra/precalc textbooks, but none of them give a satisfying theorem about this. Instead, they simply say "Make sure to check your answers, especially for solving equations involving rational polynomials or radicals".

Also, is there a rigorous theorem/proof of when extraneous roots can appear? I feel like there should be a deep and general theorem about this in algebraic geometry.

I am a first-year math grad student, and I just started teaching two sections of precalc! yikes!

So I had a major embarrasing moment during office hours when I told 2 students the wrong answer because I completely forgot about the phenomenon known as "Extraneous roots"! Does anyone remember this from grade school?

For example, if I have x^(1/4)=-2, putting both sides to the 4th power will give me x=16, but this is an incorrect solution, as 16^(1/4)=2. (EDIT: This is with the convention that roots are well-defined functions, so that the only give you the positive answer. I think if you see an equation to solve with radicals involved, it is assumed that the radicals mean positive roots). While this example is simple enough, there are cases where it is not obvious that I have an extraneous root until I plug-in to check!

I am trying to create a list of operations that cause equations to gain additional roots (or lose them). So far, I have:

Gain extraneous roots by:

1. Multiplying both sides by an expression containing a variable. (common for solving rational equations).

2. Raising both sides to an even power.

Lose roots by:

1. Dividing both sides by an expression containing a variable.

2. Taking the 2n-th square root on both sides (unless you remember to put the obligatory plus-minus sign in front).

Any others I am missing? I have consulted various algebra/precalc textbooks, but none of them give a satisfying theorem about this. Instead, they simply say "Make sure to check your answers, especially for solving equations involving rational polynomials or radicals".

Also, is there a rigorous theorem/proof of when extraneous roots can appear? I feel like there should be a deep and general theorem about this in algebraic geometry.

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