# Polynomial and Rational Functions

## Zeros of Polynomial Functions


### Learning Objectives

1. Solve quadratic and higher order equations by factoring (IA 6.5.2)

### Objective 1: Solve quadratic and higher order equations by factoring (IA 6.5.2)

In Section 5.3 we have reviewed how to solve quadratic equations by factoring. Now we will discuss how to use factoring to solve polynomial equations.
A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the highest power on any one term of the polynomial.


### Practice Makes Perfect


Solve quadratic and higher order equations by factoring.


### Practice Makes Perfect

Solve quadratic and higher order equations by factoring. 
A new bakery offers decorated, multi-tiered cakes for display and cutting at Quinceañera and wedding celebrations, as well as sheet cakes to serve most of the guests. The bakery wants the volume of a small sheet cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?
This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.

### Evaluating a Polynomial Using the Remainder Theorem

In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by 
    the remainder may be found quickly by evaluating the polynomial function at 
     that is, 
     Let’s walk through the proof of the theorem.
Recall that the Division Algorithm states that, given a polynomial dividend 
    and a non-zero polynomial divisor 
    , there exist unique polynomials 
     and 
     such that
and either 
    or the degree of 
    is less than the degree of 
   . In practice divisors, 
    will have degrees less than or equal to the degree of 
   . If the divisor, 
    is 
     this takes the form
Since the divisor 
    is linear, the remainder will be a constant, 
     And, if we evaluate this for 
     we have
In other words, 
    is the remainder obtained by dividing 
     by 
    
  

### Using the Factor Theorem to Solve a Polynomial Equation

The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm.
If 
   
   is a zero, then the remainder 
    
   is 
    
   and 
    
   or 
    
  
Notice, written in this form, 
    is a factor of 
     We can conclude if 
    
   is a zero of 
     then 
     is a factor of 
    
  
Similarly, if 
   
   is a factor of 
    
   then the remainder of the Division Algorithm 
    
   is 0. This tells us that 
    
   is a zero.
This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree 
   
   in the complex number system will have 
    
   zeros. We can use the Factor Theorem to completely factor a polynomial into the product of 
    
   factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.

### Using the Rational Zero Theorem to Find Rational Zeros

Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. But first we need a pool of rational numbers to test. The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial
Consider a quadratic function with two zeros, 
   
   and 
    
   By the Factor Theorem, these zeros have factors associated with them. Let us set each factor equal to 0, and then construct the original quadratic function absent its stretching factor.


Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4.
We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros.

### Finding the Zeros of Polynomial Functions

The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function.

### Using the Fundamental Theorem of Algebra

Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations.
Suppose 
    is a polynomial function of degree four, and 
     The Fundamental Theorem of Algebra states that there is at least one complex solution, call it 
     By the Factor Theorem, we can write 
     as a product of 
     and a polynomial quotient. Since 
     is linear, the polynomial quotient will be of degree three. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call it 
     So we can write the polynomial quotient as a product of 
     and a new polynomial quotient of degree two. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and each one will yield a factor of 
    
  

### Using the Linear Factorization Theorem to Find Polynomials with Given Zeros

A vital implication of the Fundamental Theorem of Algebra, as we stated above, is that a polynomial function of degree 
   
   will have 
     zeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function into 
     factors. The Linear Factorization Theorem tells us that a polynomial function will have the same number of factors as its degree, and that each factor will be in the form 
     where 
     is a complex number.
Let 
   
   be a polynomial function with real coefficients, and suppose 
    
   is a zero of 
    
   Then, by the Factor Theorem, 
    
   is a factor of 
    
   For 
    
   to have real coefficients, 
    
   must also be a factor of 
    
   This is true because any factor other than 
    
   when multiplied by 
    
   will leave imaginary components in the product. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. In other words, if a polynomial function 
    
   with real coefficients has a complex zero 
    
   then the complex conjugate 
    
   must also be a zero of 
     This is called the Complex Conjugate Theorem.

### Using Descartes’ Rule of Signs

There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order, Descartes’ Rule of Signs tells us of a relationship between the number of sign changes in 
    and the number of positive real zeros. For example, the polynomial function below has one sign change.
This tells us that the function must have 1 positive real zero.
There is a similar relationship between the number of sign changes in 
    and the number of negative real zeros.
In this case, 
    has 3 sign changes. This tells us that 
     could have 3 or 1 negative real zeros.

### Solving Real-World Applications 

We have now introduced a variety of tools for solving polynomial equations. Let’s use these tools to solve the bakery problem from the beginning of the section.

### Key Concepts

1. To find 
      determine the remainder of the polynomial 
       when it is divided by 
       This is known as the Remainder Theorem. See .
2. According to the Factor Theorem, 
      is a zero of 
      
     if and only if 
      
     is a factor of 
       See .
3. According to the Rational Zero Theorem, each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See  and .
4. When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
5. Synthetic division can be used to find the zeros of a polynomial function. See .
6. According to the Fundamental Theorem, every polynomial function has at least one complex zero. See .
7. Every polynomial function with degree greater than 0 has at least one complex zero.
8. Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form 
      where 
      
     is a complex number. See .
9. The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
10. The number of negative real zeros of a polynomial function is either the number of sign changes of 
     
     or less than the number of sign changes by an even integer. See .
11. Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, use the Remainder Theorem to find the remainder.
For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.
For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.
For the following exercises, find all complex solutions (real and non-real).

### Graphical

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

### Numeric

For the following exercises, list all possible rational zeros for the functions.

### Technology

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

### Extensions

For the following exercises, construct a polynomial function of least degree possible using the given information.

### Real-World Applications

For the following exercises, find the dimensions of the box described.
For the following exercises, find the dimensions of the right circular cylinder described.