### Chapter 4 Powerpoint

```College Algebra
Chapter 4
Polynomial and Rational
Functions
4.1 Polynomial Long Division and Synthetic Division
When the division cannot be completed by factoring, polynomial
long division is used and closely resembles whole number
division
In the division process, zero “place holders” are sometimes
used to ensure that like place values will “line up” as we carry
out the algorithm
Find the quotientof 8x3  27 and 2x  3
4.1 Polynomial Long Division and Synthetic Division
Find the quotientof 8x  27 and 2x  3
3
2 x  3 8 x  27
3
2 x  3 8x  0 x  0 x  27
3
2
4.1 Polynomial Long Division and Synthetic Division
 n
2

19n  2n  4  n  3
3
n  3 2n3  n 2  19n  4
4.1 Polynomial Long Division and Synthetic Division
x  4 x  21
x 3
2
x  3 x  4 x  21
2
4.1 Polynomial Long Division and Synthetic Division
If one number divides evenly into another, it must be a factor of the original number
The same idea holds for polynomials
This means that division can be used as a tool for factoring
We need to do two things first
a. Find a more efficient method for division
b. Find divisors that give a remainder of zero
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division
x
3
5

 2x 13x 17  x  5
2
1
1
-2
-13
-17
5
15
10
3
2
-7
remainder
Multiply in the diagonal direction, add in the vertical direction
Explanation of why it works is on pg 376
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division
x  12x  34x  7
x7
3
2
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division
x  15x  12
x 3
3
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division and Factorable Polynomials
Principal of Factorable Polynomials
Given a polynomial of degree n>1 with integer coefficients and
a lead coefficient of 1 or -1, the linear factors of the polynomial
must be of the form (x-p) where p is a factor of the constant
term.
Hint:
what is
easiest
Use synthetic division to help factor
x  x  10x  4 x  24
4
3
1
2
1
-10
-4
24
 1  24
 2  12
 3  8
 4  6
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division and Factorable Polynomials
x  4x  x  6
3
2
4.1 Polynomial Long Division and Synthetic Division
What values of k will make x-3 a factor of
x 2  kx  27
4.1 Polynomial Long Division and Synthetic Division
Homework pg 380 1-58
4.2 The Remainder and Factor Theorems
The Remainder Theorem
If a polynomial P(x) is divided by a linear factor (x-r), the
remainder is identical to P(r) – the original function evaluated
at r.
Use the remainder theorem to find the value of H(-5) for
H x  x  3x  8x  5x  6
4
3
2
4.2 The Remainder and Factor Theorems
Use the remainder theorem to find the value of P(1/2) for
Px  x3  2x 2  3x  2
4.2 The Remainder and Factor Theorems
The Factor Theorem
Given P(x) is a polynomial,
1. If P(r) = 0, then (x-r) is a factor of P(x).
2. If (x-r) is a factor of P(x), then P(r) = 0
Use the factor theorem to find a cubic polynomial with these three roots:
x  3, x  2, x   2
4.2 The Remainder and Factor Theorems
A polynomial P with integer coefficients has the zeros and degree
indicated. Use the factor theorem to write the function in factored
and standard form.
x  7 , x   7 , x  3, x  1; degree 4
4.2 The Remainder and Factor Theorems
Complex numbers, coefficients, and the Remainder and Factor Theorems
Show x=2i is a zero of:
Px  x  3x  4 x 12
3
2
4.2 The Remainder and Factor Theorems
Complex Conjugates Theorem
Given polynomial P(x) with real number coefficients, complex
solutions will occur in conjugate pairs.
If a+bi, b≠0, is a solution, then a-bi must also be a solution.
4.2 The Remainder and Factor Theorems
Roots of multiplicity
Some equations produce repeated roots.
Polynomial zeroes theorem
A polynomial equation of degree n has
exactly n roots, (real and complex) where
roots of multiplicity m are counted m
times.
4.2 The Remainder and Factor Theorems
Homework pg 389 1-86
4.3 The Zeroes of Polynomial Functions
The Fundamental Theorem of Algebra
Every complex polynomial of degree n≥1 has at least one complex root.
Our search for a solution will not be fruitless or
wasted, solutions for all polynomials exist.
The fundamental theorem combined with the factor theorem
enables to state the linear factorization theorem.
4.3 The Zeroes of Polynomial Functions
Linear factorization theorem
Every complex polynomial of degree n ≥ 1 can be written as the
product of a nonzero constant and exactly n linear factors
THE IMPACT
Every polynomial equation, real or complex, has exactly n
roots, counting roots of multiplicity
4.3 The Zeroes of Polynomial Functions
Find all zeroes of the complex polynomial C, given x = 1-I is a
zero. Then write C in completely factored form:
Cx  x  1  2i x  5  i x   6  6i 
3
2
4.3 The Zeroes of Polynomial Functions
The Intermediate Value Theorem (IVT)
Given f is a polynomial with real coefficients, if f(a) and f(b) have opposite
signs, there is at least one value r between a and b such that f(r)=0
HOW DOES THIS HELP???
Finding factors of polynomials
4.3 The Zeroes of Polynomial Functions
The Rational Roots Theorem (RRT)
Given a real polynomial P(x) with degree n ≥ 1 and integer
coefficients, the rational roots of P (if they exist) must be of the form
p/q, where p is a factor of the constant term and q is a factor of the
lead coefficient (p/q must be written in lowest terms)
List the possible rational roots for
3x 4  14x3  x 2  42x  24  0
4.3 The Zeroes of Polynomial Functions
Tests for 1 and -1
1. If the sum of all coefficients is zero, x = 1 is a rood and (x-1) is a factor.
2. After changing the sign of all terms with odd degree, if the sum of the
coefficients is zero, then x = -1 is a root and (x+1) is a factor.
4.3 The Zeroes of Polynomial Functions
Homework pg 403 1-106
4.4 Graphing Polynomial Functions
THE END BEHAVIOR OF A POLYNOMIAL GRAPH
If the degree of the polynomial is odd, the ends will point in opposite directions:
1. Positive lead coefficient: down on left, up on right (like y=x3)
2. Negative lead coefficient: up on left, down on right (like y=-x3)
If the degree of the polynomial is even, the ends will point in the same direction:
1. Positive lead coefficient: up on left, up on right (like y=x2)
2. Negative lead coefficient: down on left, down on right (like y=-x2)
4.4 Graphing Polynomial Functions
Attributes of polynomial graphs with roots of multiplicity
Zeroes of odd multiplicity will “cross through” the x-axis
Zeroes of even multiplicity will “bounce” off the x-axis
f x  x  3 x  2
2
Cross through
bounce
4.4 Graphing Polynomial Functions
Estimate the equation based on the graph
g(x) = (x - 2)² (x + 1)³
4.4 Graphing Polynomial Functions
Estimate the equation based on the graph
g(x) = (x - 2)² (x + 1)³ (x - 1)²
4.4 Graphing Polynomial Functions
Guidelines for Graphing Polynomial Functions
1. Determine the end behavior of the graph
2. Find the y-intercept f(0) = ?
3. Find the x-intercepts using any combination of the
rational root theorem, factor and remainder
theorems, factoring, and the quadratic formula.
4. Use the y-intercepts, end behavior, the multiplicity
of each zero, and a few mid-interval points to
sketch a smooth, continuous curve.
4.4 Graphing Polynomial Functions
Sketch the graph of
g x   x  9x  4x 12
4
2
1.
2.
3.
4.
Determine the end behavior of the graph
Find the y-intercept f(0) = ?
Find the x-intercepts using any combination
of the rational root theorem, factor and
remainder theorems, factoring, and the
Use the y-intercepts, end behavior, the
multiplicity of each zero, and a few midinterval points to sketch a smooth,
continuous curve.
Down, Down
F(0) = -12
f x  1x 1x  2 x  3
2
bounce
Cut through
Cut through
4.4 Graphing Polynomial Functions
f(x) = x⁶ - 2 x⁵ - 4 x⁴ + 8 x³
1.
2.
3.
4.
Determine the end behavior of the graph
Find the y-intercept f(0) = ?
Find the x-intercepts using any combination
of the rational root theorem, factor and
remainder theorems, factoring, and the
Use the y-intercepts, end behavior, the
multiplicity of each zero, and a few midinterval points to sketch a smooth,
continuous curve.
4.4 Graphing Polynomial Functions
Homework pg 415 1-86
4.5 Graphing Rational Functions
Vertical Asymptotes of a Rational Function
Given r x  
f x 
g x 
is a rational function in lowest
terms, vertical asymptotes will occur at the real zeroes of g
1
f x  
x2
cross
The “cross” and “bounce” concepts used for
polynomial graphs can also be applied to
rational graphs
g x  
1
x  22
bounce
4.5 Graphing Rational Functions
Given
r x  
x- and y-intercepts of a rational function
f x  is in lowest terms, and x = 0 in the domain of r,
g x 
1. To find the y-intercept, substitute 0 for x and simplify. If 0 is not in
the domain, the function has no y-intercept
2. To find the x-intercept(s), substitute 0 for f(x) and solve. If the
equation has no real zeroes, there are no x-intercepts.
Determine the x- and y-intercepts for the function
2
0
h0  2
0  3  0  10
x2
h x   2
x  3x  10
x2
0 2
x  3x  10
h0  0
0  x2
0,0
0,0
4.5 Graphing Rational Functions
Determine the x- and y-intercepts for the function
h0  
3
02  1
h0  3
Y-intercept
0,3
3
h x   2
x 1
0
3
x2 1
03
No x-intercept
4.5 Graphing Rational Functions
Given r x  
f x 
is a rational function in lowest
g x 
terms, where the lead term of f is axn and the lead term of g is bxm
Polynomial f has degree n, polynomial g has degree m
1. If n<m, the graph of h has a horizontal asymptote at y=0 (the
x-axis)
2. If n=m, the graph of h has a horizontal asymptote at y=a/b
3. If n>m, the graph of h has no horizontal asymptote
3x
r x   2
x 2
3x 2
r x   2
x 2
3x 3
r x   2
x 2
4.5 Graphing Rational Functions
Guidelines for graphing rational functions pg 428
Given r x  
f x 
is a rational function in lowest
g x 
terms, where the lead term of f is axn and the lead term of g is bxm
1.
2.
3.
4.
Find the y-intercept [evaluate r(0)]
Locate vertical asymptotes x=h [solve g(x) = 0]
Find the x-intercepts (if any) [solve f(x) = 0]
Locate the horizontal asymptote y = k (check degree of
numerator and denominator)
5. Determine if the graph will cross the horizontal asymptote
[solve r(x) = k from step 4
6. If needed, compute the value of any “mid-interval” points
needed to round-out the graph
7. Draw the asymptotes, plot the intercepts and additional
points, and use intervals where r(x) changes sign to
complete the graph
4.5 Graphing Rational Functions
3x 2  6 x  3
r x  
x2  7
1.
2.
3.
4.
5.
6.
7.
Find the y-intercept [evaluate r(0)]
Locate vertical asymptotes x=h [solve g(x) = 0]
Find the x-intercepts (if any) [solve f(x) = 0]
Locate the horizontal asymptote y = k (check degree
of numerator and denominator)
Determine if the graph will cross the horizontal
asymptote [solve r(x) = k from step 4
If needed, compute the value of any “mid-interval”
points needed to round-out the graph
Draw the asymptotes, plot the intercepts and
additional points, and use intervals where r(x)
changes sign to complete the graph
4.5 Graphing Rational Functions
Homework pg 431 1-70
4.5 Graphing Rational Functions
Given r x  
f x 
is a rational function in lowest
g x 
terms, where the lead term of f is axn and the lead term of g is bxm
Polynomial f has degree n, polynomial g has degree m
1. If n<m, the graph of h has a horizontal asymptote at y=0 (the
x-axis)
2. If n=m, the graph of h has a horizontal asymptote at y=a/b
3. If n>m, the graph of h has no horizontal asymptote
3x
r x   2
x 2
3x 2
r x   2
x 2
3x 3
r x   2
x 2
4.6 Additional Insights into Rational Functions
Oblique and nonlinear asymptotes
f x 


r
x

Given
is a rational function in lowest
g x 
terms, where the degree of f is greater than the degree of g. The
graph will have an oblique or nonlinear asymptote as determined by
f
q(x), where q(x) is the quotient of  g x

r x  
x 1
x
2

x4 1
r x   2
x
x2 1

x x
x4 1
 2
2
x
x
1
x
x
1
x  2
x
q x   x
q x   x 2
2
4.6 Additional Insights into Rational Functions
x3  4 x
v x   2
x 1
4.6 Additional Insights into Rational Functions
Choose one application problem
4.6 Additional Insights into Rational Functions
Homework pg 445 1-62
4.7 Polynomial and Rational Inequalities – An Analytical View
Solving Polynomial Inequalities
Given f(x) is a polynomial in standard form pg 452
1. Use any combination of factoring, tests for 1 and -1, the
RRT and synthetic division to write P in factored form,
noting the multiplicity of each zero.
2. Plot the zeroes on a number line (x-axis) and determine if
the graph crosses (odd multiplicity) or bounces (even
multiplicity) at each zero. Recall that complex zeroes from
irreducible quadratic factors can be ignored.
3. Use end behavior, the y-intercept, or a test point to
determine the sign of the function in a given interval, then
label all other intervals as P(x) < 0 or P(x) > 0 by
analyzing the multiplicity of neighboring zeroes.
4. State the solution using interval notation, noting
strict/non-strict inequalities.
4.7 Polynomial and Rational Inequalities – An Analytical View
f x  x  4x  3x 18,
3
2
f x   0
1.
2.
Synthetic division
x  2x2  6x  9
x  2x  32
cross
3.
4.
Use any combination of factoring, tests for 1 and -1, the
RRT and synthetic division to write P in factored form,
noting the multiplicity of each zero.
Plot the zeroes on a number line (x-axis) and determine if
the graph crosses (odd multiplicity) or bounces (even
multiplicity) at each zero. Recall that complex zeroes from
irreducible quadratic factors can be ignored.
Use end behavior, the y-intercept, or a test point to
determine the sign of the function in a given interval, then
label all other intervals as P(x) < 0 or P(x) > 0 by
analyzing the multiplicity of neighboring zeroes.
State the solution using interval notation, noting strict/nonstrict inequalities.
x   ,3   3,2
bounce
End behavior is down/up
up
down
f(x) < 0
f(x) < 0
f(x) > 0
4.7 Polynomial and Rational Inequalities – An Analytical View
x  x  5x  3  0
3
2
1.
2.
Test for 1 and -1
3.
Means that x=1 is a root
x 1x 2  2x  3
x 12 x  3
4.
Use any combination of factoring, tests for 1 and -1, the
RRT and synthetic division to write P in factored form,
noting the multiplicity of each zero.
Plot the zeroes on a number line (x-axis) and determine if
the graph crosses (odd multiplicity) or bounces (even
multiplicity) at each zero. Recall that complex zeroes from
irreducible quadratic factors can be ignored.
Use end behavior, the y-intercept, or a test point to
determine the sign of the function in a given interval, then
label all other intervals as P(x) < 0 or P(x) > 0 by
analyzing the multiplicity of neighboring zeroes.
State the solution using interval notation, noting strict/nonstrict inequalities.
x   ,3
End behavior down/up
bounce
cross
f(x) < 0
f(x) > 0
f(x) > 0
4.7 Polynomial and Rational Inequalities – An Analytical View
x
x 1 x  2

x2 x3
x 1 x  2

0
x2 x3
2
 
The graph will change signs
at x = 2, -3, and 7/4
The y-intercept is 7/6 which is positive

 4x  3  x2  4
0
x  2x  3
7

x    3,   2,  
4

4x  7
0
x  2x  3
above
above
below
below
4.7 Polynomial and Rational Inequalities – An Analytical View
above
below
above
4.7 Polynomial and Rational Inequalities – An Analytical View
Homework pg 458 1-66
Chapter 4 Review
x  2x  4
2
x x
3
3x  4
x 1 2
x x
Chapter 4 Review
Chapter 4 Review
Use synthetic division to show that (x+7) is a factor
of 2x4+13x3-6x2+9x+14
Chapter 4 Review
Factor and state roots of multiplicity
hx  x  6x  8x  6x  9
4
3
2
Chapter 4 Review
State an equation for the given graph
f x  x  1 x 1x  3
2
f x   x 4  2 x 3  4 x 2  2 x  3
Chapter 4 Review
State an equation for the given graph
x2  4x
f x   2
x 4
Chapter 4 Review
Graph
x2  9
r x   2
x  3x  4
Chapter 4 Review
Trashketball Review
Divide using long division
2x  x  8
2
x  2x
x  4 x  5x  6
x2
9x  8
2x  4  2
x  2x
x  6x  7; R  8
3
3
2
2
Chapter 4 Review
Trashketball Review
Use synthetic division to divide
x  4 x  5x  6
x2
3
2
x  6x  7; R  8
2
2 x 4  13x 3  6 x 2  9 x  14
x7
2x  x  x  2
3
2
Chapter 4 Review
Trashketball Review
Show the indicated value is a zero of the function
1
x  ; P x   4 x 3  8 x 2  3x  1
2
Chapter 4 Review
Trashketball Review
Show the indicated value is a zero of the function
x  3i; Px  x  2x  9x 18
3
2
Chapter 4 Review
Trashketball Review
Find all the zeros of the function
Real root x=3
Complex roots x=±2i
Chapter 4 Review
Trashketball Review
Find all the zeros of the function
Chapter 4 Review
Trashketball Review
State end behavior, y-intercept, and list the possible rational roots for each function
Chapter 4 Review
Trashketball Review
State end behavior, y-intercept, and list the possible rational roots for each function
Chapter 4 Review
Trashketball Review
Sketch the Graph using the degree, end behavior, x- yintercept, zeroes of multiplicity and midinterval points
Chapter 4 Review
Trashketball Review
Sketch the Graph using the degree, end behavior, x- yintercept, zeroes of multiplicity and midinterval points
Chapter 4 Review
Trashketball Review
Graph using guidelines for graphing rational functions
Chapter 4 Review
Trashketball Review
Graph using guidelines for graphing rational functions
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