Chapter 2 – Polynomial, Power, and Rational
FunctionsHW: Pg. 175 #7-16
Polynomial Functions-◦ Let n be a nonnegative integer and let a0, a1, a2,…,
an-1, an be real numbers with an≠0. The functions given by
f(x)=anxn + an-1xn-1+…+a2x2+a1x+a0
Is a polynomial function of degree n. The leading coefficient is an.
f(x)=0 is a polynomial function.*it has no degree or leading coefficient.
2.1- Linear and Quadratic Functions and Modeling
F(x) = 5x3-2x-3/4
G(x) = √(25x4+4x2)
H(x) = 4x-5+6x
K(x)=4x3+7x7
Identify degree and leading coefficient for functions:
Name Form Degree
Zero Function F(x) = 0 Undefined
Constant Function F(x)=a (a≠0) 0
Linear Function F(x)=ax+b (a≠0) 1
Quadratic Function F(x)=ax2+bx+c (a≠0) 2
Polynomial Functions of No and Low Degree
F(x) = ax+b
Slope-Intercept form of a line:
Find an equation for the linear function f such that f(-2) = 5 and f(3) = 6
Linear Functions
The average rate of change of a function y=f(x) between x=a and x=b, a≠b, is
[F(b)-F(a)]/[b-a]
Average Rate of Change
Weehawken High School bought a $50,000 building and for tax purposes are depreciating it $2000 per year over a 25-yr period using straight-line depreciation.
1. What is the rate of change of the value of the building?
2. Write an equation for the value v(t) of the building as a linear function of the time t since the building was placed in services.
3. Evaluate v(0) and v(16)4. Solve v(t)=39,000
Modeling Depreciation with a Linear Function
Point of View Characterization
Verbal Polynomial of degree 1
Algebraic F(x)=mx+b (m≠0)
Graphical Slant line with slope m and y-intercept b
Analytical Function with constant nonzero rate of change m, f is increasing if m>0, decreasing if m<0
Characteristics of Linear Functionsy=mx+b
Sketch how to transform f(x)=x2 into:
G(x)=-(1/2)x2+3
H(x)=3(x+2) 2-1
If g(x) and h(x) and in the form f(x)=ax2+bx+c, what do you notice about g(x) and h(x) when a is a certain value (negative or positive)?
Quadratic Functions and their graphs:
f(x)=ax2+bx+c
We want to find the axis of symmetry, which is x=-b/(2a).
Then:
The graph of f is a parabola with vertex (x,y), where x=-b/(2a). If a>0, the parabola opens upward, and if a<0, it opens downward.
Finding the Vertex of a Quadratic Function:
x=-b/(2a)
Use the vertex form of a quadratic function to find the vertex and axis of the graph of f(x)=8x+4x2+1:
F(x)=3x2+5x-4
G(x)=4x2+12x+4
H(x)=6x2+9x+3
f(x)=5x2+10x+5
Find the vertex of the following functions:
Any Quadratic Function f(x)=ax2+bx+c, can be written in the vertex form:
◦ F(x)=a(x-h)2+k
Where (h,k) is your vertex
h=-b/(2a) and k=is the y
Vertex Form of a Quadratic Function:
F(x)=3x2+12x+11 f(x)=a(x-h)2+k=3(x2+4x)+11 Factor 3 from the x
term
=3(x2+4x+() - () )+11 Prepare to complete the square.
=3(x2+4x+(2)2-(2)2)+11 Complete the square.
=3(x2+4x+4)-3(4)+11 Distribute the 3.
=3(x+2)2-1
Using Algebra to describe the graph of quadratic functions:
F(x)=3x2+5x-4
F(x)=8x-x2+3
G(x)=5x2+4-6x
Find vertex and axis, then rewrite functions in vertex form: f(x)=a(x-h)2+k
Characteristics of Quadratic Functions: y=ax2+bx+c
Point of View Characterization
Verbal Polynomial of degree ___
Algebraic F(x)=______________ (a≠0)
Graphical a>0
a<0
2.2 Power Functions With Modeling
HW: Pg.189 #1-10
F(x)=k*xa
◦ a is the power, k is the constant of variation
EXAMPLES:
Power function
Formulas Power Constant of Variation
C=2∏r 1 2∏
A=∏r2 2 ∏
F(x)=4x3
G(x)=1/2x6
H(x)=6x-2
F(x) = ∛x
1/(x2)
What type of Polynomials are these functions? (HINT: count the terms)
What is the power and constant of variation for the following functions:
6cx-5
h/x4
4∏r2
3*2x
ax 7x8/9
Determine if the following functions are a power function Given that a,h,and c represent constants,, and for those that are, state the power and constant of variation:
2.3 Polynomial Functions of Higher Degree
HW: Pg 203 #33-42e
F(x)=x3+x
G(x)=x3-x
H(x)=x4-x2
Find local extrema and zeros for each polynomial
Graph combinations of monomials:
F(x)=2x3
F(x)=-x3
F(x)=-2x4
F(x)=4x4
What do you notice about the limits of each function?
Graph:
F(x)=x3—2x2-15x
What do these zeros tell us about our graph?
Finding the zeros of a polynomial function:
F(x)=3x3 + 12x2 – 15x
H(x)=x2 + 3x2 – 16
G(x)=9x3 - 3x2 – 2x
K(x)=2x3 - 8x2 + 8x
F(x)=6x2 + 18x – 24
SKETCH GRAPHS:
2.4 Real Zeros of Polynomial Functions
HW: Pg. 216 #1-6
3587/32 (3x3+5x2+8x+7)/(3x+2)
Long Division
F(x) = d(x)*q(x)+r(x)
F(x) and d(x) are polynomials where q(x) is the quotient and r(x) is the remainder
Division Algorithm for Polynomials
(3x3+5x2+8x+7)/(3x+2)
Write (2x4+3x3-2)/(2x2+x+1) in fraction form
Fraction Form: F(x)/d(x)=q(x)+r(x)/d(x)
D(x)=x-k, degree is 1, so the remainder is a real number
Divide f(x)=3x2+7x-20 by:
◦ (a) x+2 (b) x-3 (c) x+5
Special Case: d(x)=x-k
Remainder Theorem:
If a polynomial f(x) is divided by x-k, then the remainder is r=f(k)
Ex: (x2+3x+5)/(x-2) k=2
So, f(k)=f(2)=(2)2+3(2)+5=15=remainder
We can find the remainder without doing long division!
Divide f(x)=3x2+7x-20 by:
◦ (a) x+2 (b) x-3 (c) x+5
Lets test the Remainder Theorem with our previous example:
If d(x)=x-k, where f(x)=(x-k)q(x) + rThen we can evaluate the polynomial f(x) at x=k:
PROVE:
F(x)=2x2-3x+1; k=2
F(x)=2x3+3x2+4x-7; k=2
F(x)=x3-x2+2x-1; k=-3
Use the Remainder Theorem to find the remainder when f(x) is divided by x-k
Now we can use this method to find both remainders and quotients for division by x-k, called synthetic division.
(2x3-3x2-5x-12)/(x-3)
K becomes zero of divisor
Synthetic Division
3 | 2 -3 -5 -12 _____________
STEPS:
* Since the leading coefficient of the dividend must be the leading coefficient , copy the first “2” into the first quotient position.
* Multiply the zero of the divisor (3) by the most recent coefficient of the quotient (2). Write the product above the line under the next term (-3).
* Add the next coefficient of the dividend to the product just found and record sum below the line in the same column.
* Repeat the “multiply” and “add” steps until the last row is completed.
(x3-5x2+3x-2)/(x+1)
(9x3+7x2-3x)/(x-10)
(5x4-3x+1)/(4-x)
Use synthetic division to solve:
Suppose f is a polynomial function of degree n1 of the form f(x)=anxn+…+a0
with every coefficient an integer. If x=p/q is a rational zero of f, where p and q have no common integer factors other than 1, then◦ P is an integer factor of the constant coefficient
a0, and
◦ Q is an integer factor of the leading coefficient an.
Rational Zero Theorem
Example: Find rational zeros of f(x)=x3-3x2+1
F(x)=3x3+4x2-5x-2
Potential Rational Zeros:
Finding the rational zeros:
F(x)=6x3-5x-1
F(x)=2x3-x2-9x+9
Find rational zeros:
Let f be a polynomial function of degree n≥1 with a positive leading coefficient. Suppose f(x) is divided by x-k using synthetic division.
If k≥0 and every number in the last line is nonnegative (positive or zero), then k is an upper bound for the real zeros of f.
If k≤0 and the numbers in the last line are alternately nonnegative and nonpositive, then k is a lower bound for the real zeros of f.
Upper and Lower Bound Tests for Real Zeros
Lets establish that all the real zeros of f(x)=2x4-7x3-8x2+14x+8 must lie in the interval [-2,5]
Example:
Now we want to find the real zeros of the polynomial function f(x)=2x4-7x3-8x2+14x+8
Establish bounds for real zeros Find the real zeros of a polynomial functions
by using the rational zeros theorem to find potential rational zeros
Use synthetic division to see which potential rational zeros are a real zero
Complete the factoring of f(x) by using synthetic division again or factor.
Steps to finding the real zeros of a polynomial function:
F(x)=10x5-3x2+x-6
Find the real zeros of a polynomial function:
F(x)=2x3-3x2-4x+6
F(x)=x3+x2-8x-6
F(x)=x4-3x3-6x2+6x+8
F(x)=2x4-7x3-2x2-7x-4
Find the real zeros of a polynomial function:
2.5 Complex Numbers
In the 17th century, mathematicians extended the definition of √(a) to include negative real numbers a.
i =√(-1) is defined as a solution of (i )2 +1=0
For any negative real number √(a) = √|a|*i
F(x)=x2+1 has no real zeros
a +bi , where a, b are real numbers
◦ a+bi is in standard form
Complex Number- is any number written in the form:
Sum: (a+bi) +(c+di) = (a+c) + (b+d)i Difference: (a+bi) – (c+di) = (a-c) + (b-d)I
EX: (a) (8 - 2i) + (5 + 4i)
(b) (4 – i) – (5 + 2i)
Sum and Difference
(2+4i)(5-i)
Z=(1/2)+(√3/2)i, find Z2
Multiply:
Z = a+bi = a – bi
When do we need to use conjugates?
Def: Complex Conjugates of the Complex Number z=a+bi is
(2+3i)/(1-5i)
Write in Standard Form:
ax2+bx+c=0
Complex Solutions of Quadratic Equations
Try: x2-5x+11=0
Solve: x2+x+1=0
Find all zeros:
f(x) = x4 + x3 + x2 + 3x - 6
DO NOW:
2.6 Complex Zeros and The Fundamental Theorem
of AlgebraHW: Pg. 234-235 #2-10e, 28-34e
Fundamental Theorem of Algebra – A polynomial function of degree n has n complex zeros (real and nonreal).
Linear Factorization Theorem – If f(x) is a polynomial function of degree n>0, then f(x) has n linear factors and
F(x) = a(x-z1)(x-z2)…(x-zn)
Where a is the leading coefficient of f(x) and z1, z2, …, zn are the complex zeros of the function.
Two Major Theorems
X=k is a…
K is a
Factor of f(x):
Fundamental Polynomial Connections in the Complex Case
F(x)=(x-2i)(x+2i)
F(x)=(x-3)(x-3)(x-i)(x+i)
Exploring Fundamental Polynomial ConnectionsWrite the polynomial function in standard form and identify the zeros :
Suppose that f(x) is a polynomial function with real coefficients. If a+bi is a zero of f(x), then the complex conjugate a-bi is also a zero of f(x)
Complex Conjugate Zeros
What can happen if the coefficients are not real?
1. Use substitution to verify that x=2i and x=-i are zeros of f(x)=x2-ix+2. Are the conjugates of 2i and –i also zeros of f(x)?
2. Use substitution to verify that x=i and x=1-i are zeros of g(x)=x2-x+(1+i). Are the conjugates of i and 1-i also zeros of g(x)?
3. What conclusions can you draw from parts 1 and 2? Do your results contradict the theorem about complex conjugates?
EXPLORATION (with your partner):
Given that -3, 4, and 2-i are zeros, find the polynomial:
Given 1, 1+2i, 1-i, find the polynomial:
Find a Polynomial from Given Zeros
Find Complex Zeros of f(x)=x5-3x4-5x3+5x2-6x+8
The complex number z=1-2i is a zero of f(x)=4x4+17x2+14x+65, find the remaining zeros, and write it in its linear factorization.
Find Complex Zeros
3x5-2x4+6x3-4x2-24x+16
Find zeros:
2.7 Graphs of Rational Functions
HW: Pg. 246 #19-30
F and g are polynomial functions with g(x)≠0. the functions:
◦ R(x)=f(x)/g(x) is a rational function
◦ Find the domain of : f(x)=1/(x+2)
Rational Functions
(a) g(x)=2/(x+3)
(b) H(x)=(3x-7)/(x-2)
Transformations: Describe how the graph of the given function can be obtained by transforming the graph f(x)=1/x
Find horizontal and vertical asymptotes of f(x)=(x2+2)/(x2+1)
Find asymptotes and intercepts of the function f(x)=x3/(x2-9)
Finding Asymptotes
Analyzing: f(x)=(2x2-2)/(x2-4)
Lets look at F(x)=(x3-3x2+3x+1)/(x-1)
2.8 Solving Equations in One Variable
HW: Pg. 254 #7-17
X + 3/x = 4
2/(x-1) + x = 5
Solve:
(2x)/(x-1) + 1/(x-3) = 2/(x2-4x+3)
3x/(x+2) + 2/(x-1) = 5/(x2+x-2)
Eliminating Extraneous Solutions:
(x-3)/x + 3/(x+2) + 6/(x2 +2x) = 0
Try:
Find the dimensions of the rectangle with minimum perimeter if its area is 200 square meters. Find the least perimeter:
Finding a Minimum Perimeter:
A = 200
2.9 Solving Inequalities in One
VariableHW: Finish 2.9 WKSH
F(x)=(x+3)(x2+1)(x-4)2
Determine the real number values of x that cause the function to be zero, positive, or negative:
Finding where a polynomial is zero, positive, or negative
(x+3)(x2+1)(x-4)2 > 0
(x+3)(x2+1)(x-4)2 ≥ 0
(x+3)(x2+1)(x-4)2 < 0
(x+3)(x2+1)(x-4)2 ≤ 0
Find solutions to:
2x3-7x2-10x+24>0
Solve Graphically: x3-6x2≤2-8x*Plug function into your calculator*
Solving a Polynomial Inequality Analytically:
Section 2.9 #1-12 odd
Check yourself
Practice: