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: