# Polynomial Functions

Textbook pages 21-24

**(i) Definition and Examples:
**

Definition:

Examples:

•

•

•

•

**(ii) Quadratic functions = Second order polynomial**

The general expression for a quadratic function is

The graph of a quadratic function is a parabola. The most common example is the
function

f(x) = x^{2}:

Examples of parabola in Nature :

The trajectory of objects thrown

• from a height (in meters)

• at velocity (in meters/second)

• at an angle α

is given by the function

where g = 9.8 m/s^{2} is the constant of gravitational
acceleration.

Example of the lucky kid shot! At what velocity is he throwing the ball?

• Height of a basketball hoop: 3.05 meters

• Angle of throw: 30°

• Length of court: 28.65 meters

**(iii) Manipulating Quadratics**

There are two typical questions one may ask:

• Type A: What is the relationship between the coefficients a , b and c and the
charac-

teristics of the graph of a parabola (position and height of the
minimum/maximum,

position of the zeros).

• Type B: Given an equation ax ^{2} + bx + c = 0, can we find x?

These questions are illustrated in this diagram:

Type A questions: To answer this question, we must
complete the square in the expression

f(x) = ax^{2} + bx + c:

That’s in the form with

Graphically, this can be interpreted as the following scaling operations on the
standard

parabola:

Type B questions: To answer this question, we start from
the result of the previous

question, and keep manipulating the expression:

so that the equation ax^{2} + bx + c = 0 becomes

with

We see that

• if D < 0 then

• if D = 0 then

• if D > 0 then

Examples:

**(iv) Factoring polynomials
**

Definition:

In practise: For quadratic polynomials f (x) = ax^{2} + bx + c

• Calculate D = b^{2}− 4ac

• Identify how many solutions there are, and what they are

• If D < 0 there are no solutions, then f(x) cannot be factored

• If D = 0 there is one solution then

• If D > 0 there are two solutions and
then

Examples:

Note: Factoring higher order polynomials is more difficult
since there exist no general

method for finding roots. However, if you know a root (for example if the root
is obvious , or

if it given to you in the problem), you can factor the polynomial using long
division .

Example:

**(v) Signs Tables
**

Signs Tables are an excellent tool to determine the sign of a polynomial function. In or-

der to use signs tables, the function must already be broken down into its factors. Then

• Draw the table

• Write

**all**the factors vertically on the left

• Write

**all**the roots horizontally on the top (in the correct order)

• Draw vertical lines below each root

• Determine and write the sign of each factor; write zeros where appropriate.

• Multiply the signs in each interval to determine the sign of the function.

Example: f(x) = 4(x − 1)(x + 2).

Example: f(x) = (x − 1)(x^{2} + 1).

Example: f(x) = (x − 2)(x + 1)(x^{2}− 3).

Check your understanding of Lecture 2

• The order of a polynomial:

What is the order of the following polynomials: f(x) = x^{4}, f(x) = 2, f(x) = (x −
2)^{7},

f(x) = (x + 2)(x^{2} + 1).

• For each of the following polynomials: (1) find the roots when they exist. (2)
write the

polynomial in factorised form. (3) draw and complete the corresponding signs
table to

identify where f(x) > 0 and f(x) < 0.

• For the first 3 polynomials of the previous question ,
determine the coordinates

of their maximum/minimum, and using all of the information you have, sketch them

on a graph.

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