Chapter 3: Polynomial Functions

Introduction

Chapter 2 the concept of functions was introduced. In this chapter we focus on polynomial functions which are a special subset of functions. Specifically, we will introduce the first and the second-degree polynomial functions of a single variable. Many real-world situations can be explained through using polynomial functions. Polynomial functions are widely used in different areas of science such as business, medicine, psychology, and sociology.

Polynomials

We shall be mainly concerned with two types of functions, namely linear and quadratic functions, which belong to a much larger class of functions called polynomials. First we give the general definition of polynomials.

A polynomial function of degree n is a function 31 of the form

31

where 31are real numbers, 31 is a natural number and 31.

The form

31

for a polynomial was first used by the French philosopher and mathematician Rene Descartes.

The degree 31 of a polynomial in one variable is the greatest exponent of its variable. The numbers

31

are called the coefficients of the polynomial. The leading coefficient 31 is the coefficient of the term with the highest degree. If 31, the polynomial function is called a monic polynomial. The number 31 is the constant coefficient or constant term.

If 31and 31, we say the degree of 31is 0. If 31, we say 31 has no degree. However, some authors prefer to say that its degree is undefined. If 31, it is called a constant function.

Polynomial equations

We have defined a polynomial function of 31 degree as a sum

31.

If 31 then its called a polynomial equation. Therefore, the polynomial equations are equations of the form

31

Suppose that 31 is a polynomial equation of degree 1. We may, hence, write it as

31

where 31 and 31 are real numbers and 31. This equation is called a linear equation in one variable.

Let f be a polynomial function. If 31 for 31, then 31 is called a root of the polynomial function. The set 31 is called a solution set.

Suppose that 31 is a polynomial equation of degree 2. We may write it as

31

where 31 and 31are real numbers and 31. This equation is called a quadratic equation in one variable.

Let us consider the quadratic function 31. We want to find a solution of 31, i.e. we want to find the number x which, when inserted in the function, will give zero. 31 means that 31. So, 31. But we know that the square of any real number is non negative. Therefore, there is no solution. For this reason, the solution set is empty set. We can show it as 31 or 31.

Suppose that 31, 31. We would like to construct a formula to find the roots of this quadratic equation. We can re-write the equation factoring out the GCF (greatest common factor).

31

We know that “31 then 31 or 31 for all 31”. So,

31

Let us use the identities given above

31

Thus,

31

Apply the square root property,

31

and

31

Finally,

31

and

31

We can write this formula (quadratic formula) in short as

31

In the quadratic formula, 31 is called the discriminant of the quadratic equation and we use the symbol 31 (capital Greek delta) for discriminant:

31

If the discriminant is positive, then there are two distinct roots. Both roots are real numbers and they are given by

31 and 31

If the discriminant is zero, then there is exactly one real root. It is sometimes called a double root. It is written as

31

If the discriminant is negative, then there are no real roots.

In general, let 31 and 31 be any two real numbers which are roots of a quadratic polynomial. This means that,

31 and 31. Thus, 31. On expanding the product, we arrive at

31.

The general form of a quadratic polynomial is

31.

If we divide both sides by 31 the polynomial becomes a monic polynomial which is

31

We observe, from the expansion above, that the sum of the roots of the polynomial is equal to 31 and their product is 31 . That means,

31

and

31.

Graphs of Polynomial Functions

Up until this part we defined polynomial functions and discussed their basic properties. We investigated the “roots” of first and second degree polynomials called linear and quadratic functions, respectively. In this part we investigate the graphs of polynomial functions. Let 31 be any polynomial function. For any real number 31, 31 is called an ordered pair.

The graph of a function 31 is the collection of all ordered pairs. The graph is drawn on the Cartesian coordinate system. So, the graph has two dimensions.

Let 31, 31 be a linear function. The graph of 31 represent a line in the plane, i.e. we use the Cartesian coordinate system to sketch the graph of 31. It has two axes which are labelled as 31and 31. From this point on we use 31 (i.e. 31).

Let 31, 31 be a quadratic function. The graph of 31 is called a parabola. In what follows we use 31 (i.e. 31). Parabolas have shapes similar to a soup bowl (see Figure 3.10). If the leading coefficient is positive, the parabola opens upward.

If the leading coefficient is negative, the parabola opens downward.

The vertex of the parabola is the lowest point on the graph if the graph opens upward and the highest point on the graph if it opens downward. The parabola is a symmetric figure and the axis of symmetry is 31 where the vertex is 31. Furthermore, 31 for 31. The 31 is a point on 31. So,

31

31

31

Thus,

31

To plot the graphs of quadratic functions, we will follow the subsequent steps.

\1. Determine whether the parabola looks up (31) or looks down (31).

\2. Determine the vertex of the parabola.

\3. Find the 31-intercept (solving 31)

\4. Find the 31-intercept (solving 31 It is clear that the 31 –intercept is 31.)

Mark the intercepts and vertex in the plane along with an extra check point, if necessary. Connect these points with a smooth curve in the shape of a bowl Proje yönetimi, projenin hedeflerine ulaşması için gerekli ihtiyaçları karşılamak üzere ilgili tüm bilgi, beceri, araç ve tekniklerin proje faaliyetlerine uygulanmasıdır. Proje yönetimi projenin fikir aşamasından kapanış aşamasına kadar geçen tüm faaliyetleri kapsar.

Polynomial Inequalities

Up until now, we have discussed the polynomial functions and equalities involving them. We now consider inequalities compromising of polynomials. A statement involving the symbols “31”, “31”, “31”, “31” is called an inequality. In this part we will focus on linear and quadratic inequalities.

We already know that any equation of the form 31 where 31, is a linear equation. Any of the following inequalities

31, 31, and 31

is called a linear inequality.

For 31, 31 defines a quadratic equation. Any of the following inequalities

31, 31, 31, and 31

is called a quadratic inequality.