PreCalc 2.5 2.5
Today you will learn about:
Zeros of a Polynomial Function
Rational Roots Theorem
PreCalc 2.5 2.5 Zeros of a Polynomial Function and Rational Roots Theorem
Fundamental Theorem of Algebra states that a Degree n polynomial has n complex zeros
first proved by German mathematician Carl Friedrich Gauss (1777­1855)
* Remember: All numbers can be written in complex format.
QUADRATIC functions have exactly 2 zeros.
f(x)=x 2­4=(x+2)(x­2)
zeros:
f(x)=x 2+4=(x+2i)(x­2i)
zeros:
The CUBIC function f(x)=x3­5x2+2x­10=(x­5)(x­√2 i)(x+√2 i) has exactly 3 zeros. zeros:
The QUARTIC function f(x)=x4­6x3+10x2­6x+9=(x­3)(x­3)(x­i)(x+i) has exactly 4 zeros zeros:
Rational Roots Theorem
(a.k.a. Rational Roots/Zeros Test)
Suppose all of the coefficients in a polynomial are integers,
and f(x) = anxn+an­1xn­1+an­2xn­2+...+a1x +a0
a0
q is a factor of an
The possible rational roots/zeros (PRR) would be
p is a factor of
p
q
Find the possible rational roots (PRR)
1) f(x)=3x3+4x2­5x­2
p( 2 ):
2) g(x)=2x4­7x3+5x2+8
p( 8 ):
q( 2 ):
q( 3 ):
p
PRR = q =
p
=
q
Note: Maximum of 4 zeros will work
PRR =
Note: Maximum of 3 zeros will work
3) h(x)=x3­3x2+1
p( 1 ):
q( 1 ):
PRR =
p
q=
Note: Maximum of 3 zeros will work
PreCalc 2.5 Find all of the rational zeros without using a calculator to graph: like 11­20
1) g(x)=3x3+4x2­5x­2
2) f(x)=2x4­7x3­26x2+23x­6
p( ):
2
q( ):
3
6
p( ):
2
q( ):
PRR:
PRR:
g(
g(
g(
g(
)= 0
)= 4
)= ­3.11
)= 0
Rational Zeros
g( 2 )= 28
g( ­2 )= 0
g( )=
g( )= 2.22
x=
Note Rational Zeros
x=
3) a. List the possible rational zeros(roots) of f.
b. Sketch the graph of f so that some of the possible
zeros can be disregarded.
c. Determine all real zeros of f (this includes rational
and irrational).
Like 25­32
a) f(x)=­3x3+20x2­36x+16
b)
p( ):
16
3
q( ):
PRR =
x=
p
q
Window:
[­2,6] x [­10,20]
c) Real Zeros x=
PreCalc 2.5 2.5 ­ DAY 2
Today we will continue learning
how to:
Find the Zeros of a
Polynomial Function
and
Write the linear factorization
of a Polynomial Function
PreCalc 2.5 4. Find all the zeros of the function and write
the polynomial as a product of linear factors
f(x)= 2x4­7x3­8x2+14x+8
Like 62­75
1. Graph and find a window
Window: [­6,8 ]x[ ­60,50]
2. Find all of the zeros with your calculator (i.e. calculate zeros) and find any rational zeros
3. use the rational zeros to synthetically divide the problem down to a quadratic
4. Factor q(x) into linear factors
5. Write all of the zeros as "x=" and write the linear factorization as "f(x)="
Answer
Zeros: x =
Factors: f(x)= (2x+1)(x­4)(x+ 2)(x­ 2)
PreCalc 2.5 7. Find all the zeros of the function and write the polynomial as a product of linear factors
a) f(x)= x2­16
Zeros:
b) f(x)= x4­16
Zeros:
x=
x=
Factors: Factors: f(x)=
For your homework,
you can use that first line where you show the factorization as your answer. Then, finish by finding the zeros. 55­61
f(x)=
Factors
Zeros
PreCalc 2.5 c) f(x)= x2­2x­1
d) f(x)= x2­10
55­61
Zeros:
x=
Factors: f(x)=
Zeros:
x=
Factors: f(x)=
e) f(x)= x2­2x+5
Zeros:
x=
Factors: f(x)=
Don't forget to write the zeros separately
PreCalc 2.5 7. Find all the zeros of the function and write the polynomial as a product of linear factors
f )
g)
­2
­4
q(x)= x2+1
=
x=
Zeros:
Zeros:
x=
Factors: Factors: f(x)=
h(x)=
1
4
PreCalc 2.5 2.5
Today you will learn about:
Zeros of a Polynomial Function
and
Linear Factorization
PreCalc 2.5 a polynomial function with real
5. Write
coefficients that has the given zeros:
a. ­2, 1+2i 1­2i b. 3, 2i Like 37­42
NOTE:
complex conjugates appear in pairs
PreCalc 2.5 6.Use given zero to find all of the zeros of the
functions
and write polynomial as a product of linear factors
Like 47­54
b. f(x) = 4x4+17x2+14x+65 a. f(x) = x4+13x2­48 zero: 1­2i
zero: 4i
1 0 13 0 ­48
4 0 17 14 65
Note:
q(x) does
not factor,
so you
must use
the
quadratic
formula
(a)
(b)
Zeros:
x=
Factors:
f(x)=