2/13/2012
Section 2.5 Zeros of Polynomial Functions
Objective 1: Use the Rational Zero Theorem to find possible Obj
i 1
h
i
l
h
fi d
ibl
rational zeros.
1
2/13/2012
Objective 2: Find zeros of a polynomial function.
Objective 3: Solve polynomial equations.
Example
List all possible rational zeros of f(x)=x3‐3x2‐4x+12
Find one of the zeros of the function using synthetic division, then factor the remaining polynomial. What are all of the zeros of the function? How can the graph below help you find the zeros?
Example
List all possible rational zeros of f(x)=6x3‐19x2+2x+3
Starting with the integers, find one zero of the function using synthetic division, then factor the remaining polynomial. What are all of the zeros of the function?
2
2/13/2012
Example
List all possible rational roots of x4‐x3+7x2‐ 9x‐18=0
Starting with the integers, find two roots of the equation using synthetic division. The graph below will help you easily find those roots. Factor the remaining polynomial. What are all of the roots of the equation? The graph below will NOT help you find the imaginary roots Why?
below will NOT help you find the imaginary roots. Why?
Always keep in mind the relationship among zeros, roots, and x‐intercepts. The zeros of a function f are the roots, or solutions, of the equation f(x) = 0. Furthermore, the real zeros, or real roots, are the x‐intercepts of the graph of f.
3
2/13/2012
Notice that the roots for our most recent problem
(x 4 -x 3 + 7 x − 9 x − 18 = 0; degree 4) were ± 3i,2,-1
Remember that having roots of 3, -2, etc. are
complex roots because 3 can be written 3+0i
and -22 can be written as -2+0i.
2+0i
4
2/13/2012
Objective 4: Use the Linear Factorization Theorem to find polynomials with given zeros.
5
2/13/2012
Example
Find a fourth‐degree polynomial function f(x) with real coefficients that has ‐1,2 and i as zeros and such that f(1)=‐ 4
Find all zeros of the polynomial function or solve the given polynomial equation.
f(x) = x3 + 12x2 + 21x + 10
6
2/13/2012
Using a graphing calculator, graph
f(x) = 2x5 + 3x4 – 5x2 – 9x – 2.
Find each of the following rounding each answer to 3 decimal p
places:
a.
b.
c.
d.
e
e.
The real zeros
The x‐intercepts
The relative maxima
The relative minima
f( 8)
f(‐8)
7