SOLUTION: I have absolutely no idea how to start because I don't really understand what the question is asking: Show that the equation x^3 + x^2 - 3 = 0 has no rational roots, but that it d

Question 174594: I have absolutely no idea how to start because I don't really understand what the question is asking:
Show that the equation x^3 + x^2 - 3 = 0 has no rational roots, but that it does have an irrational root between x=1 and x=2.
Does it mean that there is a number in between 1 and 2 that will satisfy the equation? Please help.
Answer by SAT Math Tutor(36) (Show Source): You can put this solution on YOUR website!
First, look at the factors of your first and last coefficients:
Factors of 1: +- 1
Factors of -3: +- 1, +- 3
When you have a cubic equation, any rational roots (if they exist) can be found by the quotient of the last coefficient's factors divided by the first coefficient's factors. For example, possible choices here are:
+-1, +-3
You can try those 4 and you'll find they aren't roots so there are no rational roots.
Now, consider this:
f(1) = -1
f(2) = 9
Obviously the function has a zero between 1 and 2 because the sign switches from positive to negative! We know this root isn't rational so there is an irrational root within this interval.
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