Download 2-1 Power and Radical Functions

PreCalculus
Section 2-4 Zeros of Polynomial Functions
Name: ___________________
The Rational Zero Theorem describes how the leading coefficient and constant term of a polynomial
function with integer coefficients can be used to determine a list of all possible rational zeros.
If f(x) = anxn + …+ a1x + a0, then every rational zero of f has the following form:
p
factor of constant term ( a0 )

q
factor of leading coefficient ( an )
Once you know all of the possible rational zeros of a polynomial function, you can use direct or synthetic
substitution to determine which, if any, are actual zeros of the polynomial.
Ex. 1: List all possible rational zeros of each function. Then determine which, if any, are actual zeros.
a. f  x   x3  3x2  2 x  4
b. f  x   x3  2 x  1
* When an ≠1, the list of possible zeros goes way up.
Ex. 2: List all possible rational zeros of f  x   4 x4  35x3  87 x2  56x  20 . Then determine which, if
any, are actual zeros.
You Try: List all possible rational zeros of f  x   2 x3  5x2  28x  15 . Then determine which, if any,
are actual zeros.
PreCalculus Ch 2B Notes_Page 1
Ex. 3: The water level in a bucket sitting on a patio can be modeled by f  x   x3  4x2  2x  7 where
f(x) is the height of the water in millimeters and x is the time in days. On what day(s) will the water reach
a height of 10 millimeters?
Upper and Lower Bound Tests: See key concept on page 121.
Let f (x) be a polynomial with real coefficients and a positive leading coefficient,
1. Divide f (x) by x - b (where b > 0) using synthetic division. If the last row containing the quotient and
remainder has no negative numbers, then b is an upper bound for the real roots of f (x) = 0.
---Upper bound means there is no real zeros larger than it.
2. Divide f (x) by x - a (where a < 0) using synthetic division. If the last row containing the quotient
and remainder has numbers that alternate in sign, then a is a lower bound.
---Lower bound means there is no real zeros smaller than it.
(*zero entries can be counted as a positive or a negative.)
Step 1: Graph the function to determine an interval in which the zeros lie.
Step 2: Using synthetic substitution, confirm that the upper and lower bounds by applying
the upper and lower bound test.
Step 3: Use the Rational Zero Theorem to help find all the real zeros.
Ex. 4: Determine an interval in which all real zeros of f  x   x4  4x3 11x2  4x  12 must lie. Explain
your reasoning using the upper and lower bound tests. Then find all the real zeros.
PreCalculus Ch 2B Notes_Page 2
You Try: Determine an interval in which all real zeros of f  x   x4  9x3  12x2  44x  48 must lie.
Explain your reasoning using the upper and lower bound tests. Then find all the real zeros.
Descartes’ Rule of Signs is used to narrow the search for real zeros. (See key concept on page 123.)
If f(x) = anxn + an 1xn  1 + …….+ a2x2 + a1x + a0 is a polynomial function with real coefficients, then
 The number of positive real zeros of f is equal to the number of changes in sign of
the coefficients of __________ or is less than this by some even number.

The number of negative real zeros of f is equal to the number of changes in sign of
the coefficients of __________ or is less than this by some even number.
Ex. 5: Describe the possible real zeros of f  x   x4  3x3  5x2  2x  7 using Descartes’ rule of signs.
PreCalculus Ch 2B Notes_Page 3
You Try: Describe the possible real zeros of h  x   6x5  8x2 10x  15 .
Fundamental Theorem of Algebra: See key concept on page 123.
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one solution
in the complex number system.
Corollary: If f(x) is a polynomial of degree n, then the equation f(x) = 0 has exactly n solutions including
repeated zeroes in the complex number system (Ex: repeated twice is counted as two solutions).
Linear Factorization Theorem: See key concept on page 124.
If f(x) is a polynomial of degree n where n > 0, then f(x) has precisely “n” linear factors.
f (x) = a (x – c1)( x – c2)( x – c3) … ( x – cn) where c1, c2, c3, … cn are complex numbers.
*** The leading coefficient, a, can be any nonzero real number, but to make it easier, let a = 1.
Sum and Product Theory
If p and q are zeros of f(x), then f ( x)  ( x  p)( x  q)  x 2  ( p  q) x  pq .
Ex) Find f(x) if the zeroes of f(x) are 3, -5, and 3  2 .
Ex. 6: Write a polynomial function of least degree with real coefficients in standard form that has the
following as zeros.
-1, 2, and 2 – i
PreCalculus Ch 2B Notes_Page 4
You Try: Find f(x) if the zeroes of f(x) are -3, 1 (multiplicity of 2), 4i
Factoring Polynomial Functions Over the Real numbers: See key concept on page 124.
Ex. 7:
a.
b.
c.
Consider k  x   x5  x4 13x3  23x2 14x  24 .
Write k(x) as the product of linear and irreducible quadratic factors.
Write k(x) as the product of linear factors.
List all of the zeros of k(x).
PreCalculus Ch 2B Notes_Page 5
Ex. 8: Find all complex zeros of p  x   x4  6x3  35x2  50x  58 given that 2 + 5i is a zero of p.
Then write the linear factorization of p(x).
You Try: Find all complex zeros of g  x   x5  2x4 13x3  28x2  46x  60 given that
3  i is a zero of g(x). Then write the linear factorization of g(x).
PreCalculus Ch 2B Notes_Page 6
PreCalculus
Section 2-5 Rational Functions
Name: ___________________
A rational function f(x) is the quotient of two polynomial functions a(x) and b(x), where b is nonzero.
a  x
f  x 
, b  x   0 The domain is the set of all real numbers excluding those values for b(x)=0.
b  x
Many rational functions have asymptotes, lines that the graph approaches, but does not touch.
How to find the vertical asymptotes (as x approaches a number, y approaches ± ∞)
: Set the denominator = 0 and solve.
This is like finding the bad spots in the domain. It’s where the function cannot exist.
How to find the horizontal asymptote (as x approaches ± ∞, y approaches a number)
ax n  
Let f ( x)  m
bx  
a) If n  m , then y=0 (x-axis ) is the horizontal asymptote.
3x3  1
Ex: y  4
x  3x
a
b) If n  m , then y  is the horizontal asymptote.
b
3
3x  1
Ex: y  3
x  3x
c) If n  m , then there is no horizontal asymptote. (There is a oblique asymptote if n  m  1).
3x3  1
Ex: y  2
x  3x
* You can just memorize them but it is way better to know what’s going on. What we are doing is some
quick division when x approaches ± ∞. We don’t need the remainder so we can do it in our heads.
Ex. 1: Find the domain of each function and the equations of the vertical and horizontal asymptotes, if any.
x
5 x 2  3x  1
a. f  x  
b. g ( x) 
x 1
x2
You Try: a. h  x  
2x
2
x 1
PreCalculus Ch 2B Notes_Page 7
b. k  x  
4 x2  5
2x2  x 1
To graph rational functions:
 We will need 1) x-intercepts and y-intercepts
2) vertical- and horizontal-asymptotes
 We need to remember
1) These are functions.
2) Graphs hug asymptotes.
1) How to find y-intercept (where the graph crosses the y-axis)
: Find the value of f (0) .
1.
2) How to find x-intercept (where the graph crosses the x-axis)
: Set f ( x)  0 and solve.
---since the denominator cannot be zero, you just need to set the numerator = 0.
** These are the only intercepts!
There will NOT be any others!!
See key concept on page 132 for the graphs of Rational Functions.
* To graph rational functions, simplify f if possible, and then
1. Find and sketch the asymptotes, if any
2. Find and plot the x-intercepts and y-intercept, if any
3. Find and plot at least one point in the test intervals determined by any x-int and vertical asymptotes.
Ex. 2: For each function, determine any vertical and horizontal asymptotes and intercepts. Then graph the
function and state its domain.
7
a. f  x  
x5
b. f  x  
x 1
x2  4
PreCalculus Ch 2B Notes_Page 8
You Try: Determine any vertical and horizontal asymptotes and intercepts for f  x  
x 2  x  12
.
2x2  8
Then graph the function, and state its domain.
When the degree of the numerator is exactly one more than the degree of the denominator,
the graph has a slant or oblique asymptote. See key concept on page 134.
Ex. 3: Describe any asymptotes and intercepts for f  x  
x 2x  8
. Then graph the function and state its
x3
domain.
You Try: h  x  
x 2 3x  3
x4
When the numerator and denominator have common factors, the graph has removable discontinuities
called holes, at the zeros of the common factors. See the bottom of page 135.
Ex. 5: Determine any vertical and horizontal asymptotes, holes, and intercepts for h  x  
Then graph the function and state its domain.
PreCalculus Ch 2B Notes_Page 9
x 2 9
.
x2  x  6
Rational Equations can be solved by clearing the fractions or multiplying each term in the equation by the
least common denominator. Be sure to always check for extraneous solutions ( denominator  0 ).
Ex. 6: Solve x 
4
0
x6
Ex. 7 Solve.
x
3x  2

a. x 
x 1 x 1
You Try: Solve
b.
9x
6
x2
12
2
x2


x  6x x  6
x
2
Ex. 8: The rate of the water current in a river is 4 miles per hour. In 2 hours, a boat travels 6 miles with the
current to one end of the river and 6 miles back. If r is the rate of the boat in still water, r + 4 is the rate
with the current, and r – 4 is the rate against the current. Set up the equation using d=rt and find r.
PreCalculus Ch 2B Notes_Page 10