Advanced Functions
3.3 Zeros – 3.4 Fundamental Theorem of Algebra
Name _______________________________
Date _____________ Period ________
Solutions = Roots = Zeros = x-intercepts
Example 1: Determine whether 2 is a root of: x4 – 4x3 – x2 + 4x = 0
f(2) = _______________________________ so 0 (zero)
is / is not
Example 2: Determine whether 7 is a root of:
x3 + 4x2 – 47x – 210 = 0
a root of the equation.
(Use synthetic division!)
Make 3 statements about Example 2.
1.
2.
3.
Example 3: Write the polynomial of least degree whose roots are: –7i, 7i, 4 and –4.
Example 4: Solve
(x + 3)(x2 – 16) = 0
and graph the related polynomial function.
Zeros of a Polynomial Function
Note: ALL polynomial functions have graphs that are _____________________________________ .
Multiple Zeros of a Polynomial Function:
If P(x) has (x - r) as a factor exactly k times, then r is a zero of P(x) of multiplicity k.
Example 5: Find the zeros of P(x) and state the multiplicity of each zero.
P(x) = (x – 2)2(x + 4)6(x – 8)8(3x+7)
______ occurs as a zero of multiplicity ______.
______ occurs as a zero of multiplicity ______.
______ occurs as a zero of multiplicity ______.
______ occurs as a zero of multiplicity ______.
Note: Simple Zero: _____________________________________________________________
Fundamental Theorem of Algebra
Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.
 The degree of a polynomial indicates _______________________________________________ .
 Every polynomial of degree n (n>0) can be written as ___________________________________
______________________________________________________________________________ .
P(x) =

A polynomial equation of degree n has exactly _____ complex roots.
Example 6: State the number of complex roots of each equation. Then, find the roots and graph the related
function.
a) 2x3 – 18x = 0
b) (x+4)(x–2)(x–6)(x+6) = 0
The Rational Zero Theorem:
It is important because it enables us to narrow the search for rational zeros to a finite list.
If
P(x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x1 + a0 has integer coefficients ( an  0 )
p
and
is a rational zero (in lowest terms) of P, then
q
p is a factor of the constant term a0 and q is a factor of the leading coefficient an .
Example 8: Find the roots of: x3 + 6x2 + 10x + 3 = 0
Hint: Apply the Rational Zero Theorem to find the possible rational roots!
Hint: Use synthetic division to locate a root!
Upper and Lower Bound Theorem:
Let P(x) be a polynomial function with real coefficients.
Use synthetic division to divide P(x) by (x – b), where b is a nonzero real number.

Upper Bound: (Def: No zero is greater than b.)
a) if b > 0 and the leading coefficient of P is positive,
then none of the numbers in the bottom row of synthetic division are negative!
b) if b > 0 and the leading coefficient of P is negative,
then none of the numbers in the bottom row of synthetic division are positive!

Lower Bound: (Def: No zero is less than b.)
a) if b < 0 and the numbers in the bottom row of synthetic division alternate in sign!
 Note: the number “0” can be considered positive or negative as needed to produce an alternating sign
pattern.
Example 9: Use the Upper- and Lower-Bound Theorem to locate the smallest positive integer that is an
upper-bound and the largest negative integer that is a lower-bound of the real zeros of
P(x) = 2x3 + 7x2 – 4x – 14
Hint: Apply the Rational Root Theorem to find the possible rational roots!
Hint: For the Upper-bound, try x = 1, x = 2, …
Hint: For the Lower-Bound, try x = –1, x = –2, …
Zero Location Theorem:
Let P(x) be a polynomial function and let a and b be two distinct real numbers.
If P(a) and P(b) have opposite signs, then there is at least one real number c
between a and b such that P(c) = 0.
Example 10: Use the Zero Location Theorem to state between which two consecutive integers the real zeros
of P(x) = 2x3 + 7x2 – 4x – 14 are located.
x
y
Real Zeros occur between _______ and _________
Real Zeros occur between _______ and _________
Real Zeros occur between _______ and _________
Use the calculator to locate the zeros of P(x) = 2x3 + 7x2 – 4x – 14
Sketch the graph below.
to the nearest tenth.
HW: p. 316-317 #1-4, 7-10, 17, 22, (37-39, 50-51 show synthetic)
p. 327 # 1-2, 11-19 odd, 31-34
*** Test Chapter 3.1- 3.4 in 2 classes (Nov15-16)