Question 465494: Given the polynomial f(x) = 2x4 - 18x2
a. Use Descartes Rule of Signs to determine the number of positive and negative roots.
b. Use the Rational Zero Theorem (aka Rational Roots Theorem) to determine a list of possible zeros.
c. Use the Intermediate Value Theorem to prove that the polynomial has a zero in the interval [-6,-1].
d. Solve for the zeros of f(x).
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! 
If x = 0 (up to multiplicity) is an obvious root of the polynomial, then factor out the highest power of x from the expression, and then apply the relevant theorem. Hence  , and apply simply look at  .
a. There is only one variation of sign among the terms of the polynomial, and so there is one positive real root. If we substitute -x for x in the polynomial, we get the same function, and so this tells us that there is also one negative root.
b. From the rational roots theorem, the possible rational roots of are simply the divisors of 9, namely -3, -1,1, and 3.
c. Using ,  , while  , hence by the intermediate value theorem, there is r between (-6, -1) such that f(r) = 0.
d. 
==> x = 0,0,-3,3 are the roots.
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