Question 87746: Use the rational zeros theorem and the equation x4 – 12 = 0 to show that (12)¼ (i.e. the 4th root of 12) is irrational.
Answer by longjonsilver(2297)   (Show Source):
You can put this solution on YOUR website! i have never used this rational zeros theorem - i have had a quick look on the web and i think i understand, so here goes.
q is the factors of 1 --> the coefficient of the x^4 term
p is the factors of 12
q = +1, -1
p = 1,2,3,4,6,12,-1,-2,-3,-4,-6,-12
Now, N=p/q gives all RATIONAL values that when put into the polynomial and give zero are solutions. For our values of p and q, we get N=1,2,3,4,6,12,-1,-2,-3,-4,-6,-12 as possible solutions to the polynomial.
However, putting each of these as x into the polynomial - none gives us zero when we raise it to the power 4 and then subtract 12.
So, the conclusion is that if there is a real solution, it has to be irrational. Thinking about this particular case, we have  and so there is a real solution...there is a number that multiplies by itself 4 times to give 12 and it is between 1 and 2 since  and  .
cheers
Jon.
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