• UNIT 7 : Polynomial, rational and irrational functions

    Key unit competence 

    Use concepts and definitions of functions to determine the domain of rational functions and represent them graphically in simple cases and solve related problems. 

    Learning objectives 



    7.1 Polynomials

    Activity 7.1 

    Research on the definition of polynomial. Discuss with the rest of the class to come up with a concise definition.

    Polynomial 

    comes from poly- meaning “many” and -nomial meaning “term”.  A polynomial looks like this: 

            

    A polynomial can have: 

    • constants (like 3, –20, or ½) 

    • variables (like x and y) 

    • exponents (like the 2 in y2), but only 0, 1, 2, 3, ... etc 

    that can be combined using addition, subtraction, multiplication and division.

    A polynomial can never be divided by a variable such as 

    Generally  




    • Polynomials can have no variable at all: For example, 23 is a polynomial. It has just one term, which is a constant. 

    • Or one variable: For example,  x4 – 2x2+x has three terms, but only one variable (x) 

    • Or two or more variables: For example: xy4 – 5x2z has two terms, and three variables (x, y and z) 

    The degree of a polynomial with only one variable is the largest exponent of that variable. For example, in 4x3 – x + 3 the degree is 3 the largest exponent of x.


    Factorization of polynomials

    Activity 7.2 

    In groups, discuss the meaning of factorization. What are the different operations involved in factorization?

    In arithmetic, you are familiar with factorization of integers into prime factors. 

    For example, 6 = 2 × 3. 

    The 6 is called the multiple, while 2 and 3 are called its divisors or factors. The process of writing 6 as product of 2 and 3 is called factorizationFactors 2 and 3 cannot be further reduced into other factors. 

    Like factorization of integers in arithmetic, we have factorization of polynomials into other irreducible polynomials in algebra. 

    For example, x2 + 2x is a polynomial. It can be factorized into x and (x + 2). x2 + 2x = x (x + 2). 

    So, x and x + 2 are two factors of x2 + 2x. While x is a monomial factor,  x + 2 is a binomial factor. 

    Factorization is a way of writing a polynomial as a product of two or more factors. Here is an example: 

    x3 – 3x2 – 2x + 6 = (x – 3) (x2 – 2) = 


    Types of factorization

    1. Factorization into monomials 

    Remember the distributive property? It is  a (b + c ) = ab + ac


    2. By grouping of terms




    3. Factorization of trinomial perfect squares 

    Trinomial perfect squares have three monomials, in which two terms are perfect squares and one term is the product of the square roots of the two terms which are perfect squares 



    4. Factorization of trinomials of the form x2 + bx + c




    5. Factorization of trinomials of the form ax 2 + bx + c



    6. Factorization of difference of two perfect squares



    Roots of polynomials 

    We say that x = a is a root or zero of a polynomial P(x) if it is a solution to the equation P(x) = 0. In other words, x = a is a root of a polynomial P(x) if P(a) = 0. In that case x – a is one of the factors of the polynomial P(x).



    7.2 Numerical functions

    Definition 

    Consider two sets A and B. A function from A into B is a rule which assigns a unique element of A exactly one element of B. We write the function from A to B as f: A → B. Most of time we do not indicate the two sets and we simply write y = f(x). 

    The sets A and B are called the domain and range of f, respectively. The domain of f is denoted by dom (f).




    Domain and range of a function 

    We have been introduced to functions. These involve the relationship or rule connecting domains to ranges. 


    Note that we have to check whether the relation is a function by looking for duplicate x-values. If you find a duplicate x-value, then the different y-values mean that you do not have a function.

    Domain 

    Domain of a function is the set of all real numbers for which the expression of the function is defined as a real number. In other words, it is all the real numbers for which the expression makes sense. 

    Determination of domain of definition of a function in lR

    When determining domain of definition, we have to take into account three things:

    • You cannot have a zero in the denominator. 

    • You cannot have a negative number under an even root. 

    • You cannot evaluate the logarithm of a negative number


    Range 

    Let f: A → B be a function. The range of f, denoted by Im(f) is the image of A under f, that is, Im (f) = f[A]. The range consists of all possible values the function f can have. 


    Steps to finding range of function 

    To find the range of function f described by formula where the domain is taken to be the natural domain: 

    1. Put y = f(x). 

    2. Solve x in terms of y 

    3. The range of f is the set of all real numbers y such that x can be solved.


       


    As we can see from, the graph “covers” all y-values (that is, the graph will go as low as I like, and will also go as high as I like). Since the graph will eventually cover all possible values of y, then the range is “all real numbers”.

       

       

     

       


    Composition of functions 

    The composition of g and f, denoted g f is defined by the rule (g f)(x) = g(f(x))  provided that f(x) is in the domain of g. 


     

    The inverse of a numerical function 

    For a one-to-one function defined by y = f(x), the equation of the inverse can be found as follows: 

    I        Replace f(x) by y. 

    II       Interchange x and y. 

    III      Solve for y. 

    IV      Replace y by f–1(x)





    Parity of a function 

    Even function 




    Odd function 

    We say that a function f is odd if ∀x ∈ Dom (f), (–x) ∈Dom(f); 

    f(–x) = –f(x) 

    For example: 




    7.3 Application of rational and irrational functions

    Activity 7.3 

    In groups of five, research on the different ways of applying functions of rational and irrational number. Discuss your findings in class. Have at least one person from each group demonstrate an example of application to the rest of the class.

    Free falling objects 

    A free falling object is accelerated toward the earth with a constant acceleration equal to g = 9.8 m/s2. The different formulas used in dealing with free falling objects are: 





    Projectiles 

    We consider the motion of a projectile fired from ground level at an angle q upward from the horizontal and an initial speed v0. We are interested in finding, how long it takes the projectile to reach the maximum height, the maximum height the projectile reaches and the horizontal distance it travels (called the range, R). In general we use the following formulas: 




    Determination of the equilibrium cost (price) of a commodity in an isolated market 






    Rates of reaction in chemistry 

    The rate of reaction of a chemical reaction is the rate at which products are formed. To measure a rate of reaction, we simply need to examine the concentration of one of the products as a function of time.




    UNIT 6 : Quadratic equations and inequalitiesUNIT 8 : Limits of polynomial, rational and irrational functions