3.6
The Real Zeros of Polynomial Functions
Goals:
• Finding zeros of polynomials
• Factoring polynomials completely
Review
19

4
dividend
remainder
 quotient 
divisor
divisor
Review : Synthetic Division
f ( x)  2 x 3  x 2  2 x  3
f ( x)

x 1
f ( x)

x 1
f ( x)  d ( x)q ( x)  r ( x)
2. Remainder Theorem
Remainder Theorem.
For any polynomial f(x) the remainder of
𝑓 𝑥
𝑥−𝑐
is the number 𝑓 (𝑐)
2x  x  2x  3
Determine the remainder of :
x 1
3
2
3. Application of Remainder Theorem
1) Determine the remainder of
2 x 35  2 x18
x 1
2) Determine the remainder of 2 x89  x 2
x 1
4. Recall: Factor Theorem
𝑓 𝑐 =0
if and only if
(𝑥 − 𝑐) is a factor of 𝑓 𝑥 .
5. Application of Factor Theorem
f ( x)  2 x 22  4 x8  x 5  2 x18  1
1) Is x + 1 a factor of f (x ) ?
2) Is x - 1 a factor of f (x ) ?
6. Factoring of Polynomials
If (𝑥 − 𝑐) is a factor of 𝑓
then f ( x)  ( x  c)  ( quotient )
Is 𝑥 − 1 a factor of
f ( x)  x 4  6 x 3  x 2  26 x  20
If yes, then write f(x) in factored form:
f ( x)  ( x  c)  (quotient )
 ( x  1)( x3  7 x 2  6 x  20)
?
summary
If -3 is a zero of f (x) . What does the factor
theorem tell us?
1. f (3)  0
2. 𝑥 + 3 is a factor of 𝑓(𝑥).
3. The remainder of
𝑓(𝑥)
(𝑥+3)
is zero
4. The point (-3,0) is an x-intercept on the graph.
6. Real zeros of a polynomial
Number of Real Zeros Theorem
A polynomial of degree n, has at most n real zeros.
Types of Zeros:
Example of a factored polynomial:
f ( x)  ( x  3)( 2 x  1)( x  2 )( x  4  5i )( x  4  5i )
Rational Zeros
Irrational Zeros
Complex Imaginary Zeros
6. Real zeros of a polynomial
2 Methods for finding the zeros
1) Graphing calculator
(gives approximation to irrational zeros)
2) Algebraically
(better for finding exact value of zeros)
6 a) graphing calculator approx.
Find zeros (x-intercepts) using graphing calculator.
Graph:
p. 184 #81.
x-intercepts: Use ZERO feature
y-intercepts: TRACE: x=0
d) Table to determine graph close to
zero. Is it above or below?
e) Max/Min
6b) Identify Rational Zeros
Rational Roots Theorem
Given: a polynomial
f (x) with integer coefficients.
If 𝑓(𝑥) has any rational zeros, they will be from the list:
 p
 
q
where p = factors of constant term
q = factors of leading coefficient
3
2
f
(
x
)

x

14
x
 29 x  44
List all possible rational zeros.
8. Test a potential zero
1) graphing calculator (TABLE or Trace)
OR
2) Does f(c) = 0 ?
f ( x)  x 3  14 x 2  29 x  44
9. Determine the zeros of a polynomial
Finding both rational and irrational zeros.
1)
2)
3)
4)
Find zeros on calculator and verify f(c) = 0
How many zeros (x-intercepts) are there?
Are any zeros repeated?
Continue synthetic division on previous solution until quotient
is factorable.
Example 1 : f ( x)  4 x  8x  x  2
5
4
Example 2 : f ( x)  x  6 x  17 x  28x  20
4
3
2
10. Write the complete factorization
Write as product of:
•linear factors
and
•irreducible quadratic factors
Example: This function is completely factored


2 2

f ( x)  ( x  1)  x   x  1
3

2
10. Write the complete factorization
Example 3:
f ( x)  x 4  6 x 3  x 2  26 x  20
f ( x)  ( x  c1 )( x  c2 )( x  cn )(irreducibl e part )
1) Find rational zeros
2) Perform synthetic division on each quotient
3) Repeat until reduced to easily factorable
quotient .
6. Write the complete factorization
Example 4 : f ( x)  2 x  3x  31x  24 x  20
4
3
2
Look for repeated zeros (where graph touches at the zero)
6. Write the complete factorization
Example 5 : f ( x)  x  x  6 x  4 x  4 x  24
6
5
4
2
Reduces to difference of squares that can be factored.
6. Write the complete factorization
Example 6 : f ( x)  3x  x  15x  5
3
2
If integer zeros are not found on calculator, look for zeros
from list of potential zeros. {p/q} and verify.