POLYNOMIAL
FUNCTIONS
Problems
Problem : How many variations
are in the function f (x) = x 6 -6x 4 +2x 2 +
2x - 7 ?
Problem : How many positive and
negative real roots might the function f (x) = 2x 3-3x 2 -
4x + 1 have?
f (x) has two variations, so
it may have two or zero positive roots. f (- x) has
one variation, so it has one negative root.
Problem : What are the possible
rational roots of f (x) = x 3 -4x 2 +
3x - 6 ? Use Descartes' Rule of Signs and the Rational Root
Theorem.
Descartes' Rule of Signs indicates that there
are either three or one positive real roots, and no negative real roots. The
Rational Root Theorem indicates that the possible rational roots
are ±1 , ±2 , ±3 , and ±6 . Knowing
that there are no negative roots, the possible rational roots
are 1 , 2 , 3 , and 6 .
Problem : Find all real roots
of x 3 +
x 2 - 4x -
2 .
x = { -2, -
, 2} .
Problem : Find all real roots
of x 4 +5x 3 - 15x +
9 .
x = { -3, 1, -3.7912…, 0.7912…}
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