First we'll develop a polynomial equation with
solutions 1,1,-5/2, and -1.
x = 1, x = 1, x = -5/2, x = -1
x-1 = 0 x-1 = 0 x+5/2 = 0 x+1 = 0
(x-1)(x-1)(x+5/2)(x+1) = 0
To avoid fractions multiply through by 2
(x-1)(x-1)(2)(x+5/2)(x+1) = 0
(x-1)(x-1)(2x+5)(x+1) = 0
(x²-2x+1)(2x²+7x+1) = 0
2x4+3x³-7x²-3x+5 = 0
That's a quartic equation with those roots.
Therefore a quartic function with those zeros is:
p(x) = 2x4+3x³-7x²-3x+5
To graph it
1. the coefficient of the leading term 2x4 is positive
and therefor the graph goes upward on the extreme right side.
2. the degree is 4, which is an even number, so the graph
also goes up on the extreme left.
The first zero on the left is -5/2 or -2.5, so the curve
comes downard from the left and since -2.5 has odd multiplicity,
1, it cuts through the x-axis at -2.5 going downward to the right.
2. The next zero is -1, so the graph must turn around and go
upward. Since -1 has odd multiplicity, 1, it cuts through the
x-axis at -1 going upward to the right.
3. The next and final zero is 1, so the graph must turn around and go
downward. Since 1 has even multiplicity, 2, the graph bounces off
the x-axis at 1 going back upward to the right.
You might plot these points:
(-2.6,4.1), (-2.5,0), (-2,-9), (-1.3,-3.8), (-1,0), (0,5), (.5,2.25),
(1,0),(1.5,5)
Edwin
