Chapter 2.3: Real Zeros of Polynomial Functions

There are Two Types of ways to divide Polynomial Functions:

• Long Division:

This method usually takes a lot longer and fills up a lot of space. Most people prefer to use the second method when possible. However, in all cases long division will work when Synthetic will not.

• Synthetic Division:

Rules for Synthetic Division: 

• To use Synthetic Division, the factor that is dividing the polynomial must be to the factor of 1, no larger, no smaller.
• The remainder must be added to the end of the answer above the factor used for dividing

ex.

Remainder Theorem: 

• When using Synthetic Division, if you plug in a number that results in a remainder, the number and the reminder is a point on the graph. of the polynomial being divided.
• For the equation, , when the number, 1 or (x-1), is used in synthetic division, a remainder of 1 is left over. We can then say that the point (1,1) is on the graph of.

Rational Zero Test:

• In a polynomial, take the constant and divide it by the leading coefficient to find all the possible x-intercepts.
• For the equation,, all zeros can be found by taking every number from the following list.. and putting them through synthetic division to see which ones produce a remainder of zero. 
• If the numbers in a equation add up to zero then the number, 1, will work.

Chapter 2.2: End Behavior of Polynomial Functions

End Behavior: The Behavior of a graph as the graph of f(x) as x approaches infinity or negative infinity.

Polynomial functions with even degrees (ex.  x²+8x-15) approach either infinity or negative infinity on both ends of the graph, but never both.

Polynomials with odd degrees (ex. 2x³+5x²-13x+12) approach infinity on one end of the graph and negative infinity on the other. 

• Whether the function with the largest degree is positive or negative determines which way the ends of the graph will behave. 
• If a function with an even degree is negative, the ends will both approach negative infinity.

Example: Graph of f(x)=-x²-6x+8

• If a function with an odd degree is negative, the graph will approach infinity on the left side and approach negative infinity on the right.

The reason that the first degree determines the behavior of the graph is because, during End Behavior, when x approaches infinity or negative infinity, it overshadows all lesser values due to having the largest exponent.

As \[x\rightarrow \infty\] or -∞, f(x) approaches infinity as well.

Multiplicity

The zeroes of a graph of a polynomial function correspond to x-intercepts. Each intercept has a "multiplicity", or how many times it's associated factor appears in the polynomial. 

Ex. f(x)=(x+3)(x-2) has multiplicities of one -3 and one 2. f(x)=(x+3)(x-2)² has multiplicities of one -3 and two 2.

• The graph of function (x+6)(x-7) has 7 multiplicity 1 and -6 multiplicity 1.

• The graph of function (x+6)(x-7)(x-7) has -6 multiplicity 1 and 7 multiplicity 2.

Both functions have the same x-intercepts but have different multiplicities. The number of occurrences of the intercepts determine whether the line crosses the x-axis or turns away from it.

If the multiplicity is greater than 1, it will turn back.

2.1 Polynomial and Quadratic Functions

Polynomial Function:

Let n be a nonnegative integer and let a_n, a_n-1, . . . , a₂, a₁, a₀ be real numbers with a_n ≠ 0.

The function f(x) = a_nxⁿ + a_n-1x^n-1 + . . . + a₂x² + a₁x + a₀  is called a polynomial function of x with degree n.

Polynomial functions are classified by degree (the highest value for n where a_n is not equal to 0)

·      Constant function

o  f(x) = a, a ≠ 0;

o  Has a degree 0

·      Linear function

o  f(x) = mx + b, m ≠ 0

o  Has degree 1

Quadratic Function

·      Definition: Let a, b, and c b real number with a ≠ 0. Has degree 2. The function is f(x) = ax² + bx + c

·      Graph is a parabola, a special type of U-shaped curve

o  All parabolas are symmetric with respect to the axis of symmetry (axis of parabola)

·      Vertex is the point where the axis intersects the parabola

o  If a > 0, the vertex is the minimum point on the graph and the parabola opens upwards

o  If a < 0, the vertex is the maximum point on the graph and the parabola opens downwards

Standard Form of a Quadratic Function

·      Standard Form is f(x) = a(x-h)² + k

o   Especially convenient for sketching a parabola because it identifies vertex of the parabola as (h, k)

·      Identifying the Vertex of a Quadratic Function:

o  Describe the graph of f(x) = 2x² + 8x + 7 and identify the vertex

o  Solution:

§  f(x) = 2x² + 8x + 7

                = 2(x² + 4x) + 7

                = 2(x² + 4x + 4 – 4) + 7

·      Because b = 4, add and subtract (b/2)² = 4 within parentheses               

                = 2(x² + 4x + 4) – 2(4) + 7

                = 2(x + 2)² – 1

§  Vertex is (-2, -1)

1.5 Inverse Functions

Definition of the Inverse of a Function

Let f and g be two functions such that
f(g(x))=x          for every x in the domain of g

and

g(f(x))=x          For every x in the domain of f

Under these conditions, the function g is the inverse of the function f. The function g is denoted by , and the range of f must be equal to the domain of .

Expressing an inverse function

An inverse function is expressed by the term , It is not referring to the reciprocal of f(x), which is 

Finding the Inverse of Functions Informally
Ex. Find the inverse of f(x)=4x. Then verify that both  and  are equal to the identity function.

Solution
The given function multiplies each input by 4 "undo" this function. you need to divide each input by 4. So, the inverse function of f(x)=4x is
 


Finding the Inverse of a Function
1. Use the Horizontal line test to decide whether f has an inverse.
2. In the equation for f(x), replace f(x) by y.
3. Interchange the roles of x and y, and solve for y
4. Replace y by (x) in the new equation.
5. verify that f and  are inverses of each other by showing that  = x and 


Ex. Find the inverse (if it exists) of 







2x= 5 - 3y

3y = 5 - 2x

1.4 Compositions of Functions

Compositions of Functions is another way of combining two functions. When these two functions form together, they create the composition of one with the other.

The composition of the functions f with g is

 

 

Example 1: Forming the Composition of f with g

• Find the composition function for the following:

 

Example 2: Forming the Composition of g with f

• Find the composition function for the following:





 

Example 3:

• Find the composition function for the following: 
   
  
• If possible, find
  
  

The domain of the composition function is [1, ∞].

So,

is defined, but

is not defined because 0 is not in the domain of

 

 

Example 4: A Case in Which

• Given
   
  
• Find the following:
  a. 
  b.

a.                                                        

 

 

b.

 

So,

 but this isn't always true. There are cases where the two do not equal each other.

 

Practice Problems

 

1. Given

 

Find

a.

b.

 

Answer:

 a.

b.

 

2. Given

 

Find

a.

b.

 

Answer:

a.

b.