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College AlgebraChapter 4
Polynomial and RationalFunctions
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
32278 3 xandxofquotienttheFind
32278 3 xandxofquotienttheFind
4.1 Polynomial Long Division and Synthetic Division
2700832 23 xxxx
27832 3 xx
4.1 Polynomial Long Division and Synthetic Division
34219 32 nnnn
41923 23 nnnn
4.1 Polynomial Long Division and Synthetic Division
3
2142
x
xx
2143 2 xxx
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 divisionb. Find divisors that give a remainder of zero
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division
517132 23 xxxx
1 -2 -13 -175
Multiply in the diagonal direction, add in the vertical direction
1
5
3
15
2
10
-7
remainder
Explanation of why it works is on pg 376
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division
7
73412 23
x
xxx
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division
3
12153
x
xx
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division and Factorable Polynomials
Principal of Factorable PolynomialsGiven 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.
24410 234 xxxxUse synthetic division to help factor
64
83
122
241
Hint: Start with
what is easiest
1 1 -10 -4 24
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division and Factorable Polynomials
64 23 xxx
4.1 Polynomial Long Division and Synthetic Division
What values of k will make x-3 a factor of 272 kxx
4.1 Polynomial Long Division and Synthetic Division
Homework pg 380 1-58
4.2 The Remainder and Factor Theorems
The Remainder TheoremIf 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
6583 234 xxxxxH
4.2 The Remainder and Factor Theorems
Use the remainder theorem to find the value of P(1/2) for
232 23 xxxxP
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:
2,2,3 xxx
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.
4 degree;1,3,7,7 xxxx
4.2 The Remainder and Factor Theorems
Complex numbers, coefficients, and the Remainder and Factor Theorems
Show x=2i is a zero of:
1243 23 xxxxP
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 multiplicitySome 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:
ixixixxC 66521 23
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 02442143 234 xxxx
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:3. Positive lead coefficient: up on left, up on right (like y=x2)
4. 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-axisZeroes of even multiplicity will “bounce” off the x-axis
23 2 xxxf
bounce
Cross through
4.4 Graphing Polynomial Functions
Estimate the equation based on the graph
1)³ +(x 2)² -(x = g(x)
4.4 Graphing Polynomial Functions
g(x) = (x - 2)² (x + 1)³ (x - 1)²
Estimate the equation based on the graph
4.4 Graphing Polynomial Functions
Guidelines for Graphing Polynomial Functions
1. Determine the end behavior of the graph2. 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
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.
Sketch the graph of 1249 24 xxxxg
Down, Down
F(0) = -12
3211 2 xxxxfCut through
Cut through
bounce
4.4 Graphing Polynomial Functions
f(x) = x⁶ - 2 x⁵ - 4 x⁴ + 8 x³ 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
Homework pg 415 1-86
4.5 Graphing Rational Functions
Vertical Asymptotes of a Rational Function
Given is a rational function in lowest
terms, vertical asymptotes will occur at the real zeroes of g
xgxf
xr
The “cross” and “bounce” concepts used for polynomial graphs can also be
applied to rational graphs 2
1
x
xf 22
1
x
xg
bouncecross
4.5 Graphing Rational Functions
x- and y-intercepts of a rational functionGiven is in lowest terms, and x = 0 in the domain of r,
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.
xgxf
xr
Determine the x- and y-intercepts for the function 1032
2
xx
xxh
10030
00
2
2
h 103
02
2
xx
x
20 x 00 h
0,0 0,0
4.5 Graphing Rational Functions
Determine the x- and y-intercepts for the function 1
32
x
xh
1
30
2 x
10
30
2 h
30
No x-intercept
30 h
3,0
Y-intercept
4.5 Graphing Rational Functions
Given is a rational function in lowest
terms, where the lead term of f is axn and the lead term of g is bxm
xgxf
xr
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 (the ratio of lead coefficients)
3. If n>m, the graph of h has no horizontal asymptote
2
32
x
xxr
2
32
2
x
xxr
2
32
3
x
xxr
4.5 Graphing Rational Functions
Guidelines for graphing rational functions pg 428
1. Find the y-intercept [evaluate r(0)]2. Locate vertical asymptotes x=h [solve g(x) = 0]3. Find the x-intercepts (if any) [solve f(x) = 0]4. 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 46. If needed, compute the value of any “mid-interval”
points needed to round-out the graph7. Draw the asymptotes, plot the intercepts and
additional points, and use intervals where r(x) changes sign to complete the graph
Given is a rational function in lowest
terms, where the lead term of f is axn and the lead term of g is bxm
xgxf
xr
4.5 Graphing Rational Functions1. Find the y-intercept [evaluate r(0)]2. Locate vertical asymptotes x=h [solve g(x) =
0]3. Find the x-intercepts (if any) [solve f(x) = 0]4. 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
7
3632
2
x
xxxr
4.5 Graphing Rational Functions
Homework pg 431 1-70
4.5 Graphing Rational Functions
Given is a rational function in lowest
terms, where the lead term of f is axn and the lead term of g is bxm
xgxf
xr
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 (the ratio of lead coefficients)
3. If n>m, the graph of h has no horizontal asymptote
2
32
x
xxr
2
32
2
x
xxr
2
32
3
x
xxr
4.6 Additional Insights into Rational Functions
Oblique and nonlinear asymptotes
Given is a rational function in lowest
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 q(x), where q(x) is the quotient of
xgxf
xr
xg
f
x
xxr
12
xx
x 12
xx
1
xxq
2
4 1
x
xxr
22
4 1
xx
x
22 1
xx
2xxq
4.6 Additional Insights into Rational Functions
1
42
3
x
xxxv
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 4521. 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
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.
0,1834 23 xfxxxxf
Synthetic division
962 2 xxx
232 xx
bouncecross
End behavior is down/up
down
up
f(x) > 0f(x) < 0f(x) < 0
2,33, x
4.7 Polynomial and Rational Inequalities – An Analytical View
03523 xxx1. 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.
Test for 1 and -1
Add coefficients 1+1+-5+3=0Means that x=1 is a root
31 2 xx
cross bounce
End behavior down/up
f(x) < 0 f(x) > 0 f(x) > 0
3,x
321 2 xxx
4.7 Polynomial and Rational Inequalities – An Analytical View
3
2
2
1
x
x
x
x
03
2
2
1
x
x
x
x
0
32
434 22
xx
xxx
032
74
xx
x
The graph will change signs at x = 2, -3, and
7/4
The y-intercept is 7/6 which is positive
above
belowbelow
above
,2
4
7,3x
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
xx
xx
2
3 42xx
xx
2
431
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
9686 234 xxxxxh
Chapter 4 Review
State an equation for the given graph
311 2 xxxxf
3242 234 xxxxxf
Chapter 4 Review
State an equation for the given graph
4
42
2
x
xxxf
Chapter 4 Review
Graph 43
92
2
xx
xxr
Chapter 4 Review
Trashketball Review
Divide using long division
xx
xx
2
822
3
xx
xx
2
8942
2
2
654 23
x
xxx
8;762 Rxx
Chapter 4 Review
Trashketball Review
Use synthetic division to divide
2
654 23
x
xxx
8;762 Rxx
7
1496132 234
x
xxxx
22 23 xxx
Chapter 4 Review
Trashketball Review
Show the indicated value is a zero of the function
1384;2
1 23 xxxxPx
Chapter 4 Review
Trashketball Review
Show the indicated value is a zero of the function
1892;3 23 xxxxPix
Chapter 4 Review
Trashketball Review
Find all the zeros of the function
Real root x=3Complex 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- y-intercept, zeroes of multiplicity and midinterval points
Chapter 4 Review
Trashketball Review
Sketch the Graph using the degree, end behavior, x- y-intercept, 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