Polynomial Functions

Textbook pages 21-24

(i) Definition and 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) = x2:

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/s2 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) = ax2 + bx + c:

That’s in the form with

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

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 ax2 + bx + c = 0 becomes


We see that
• if D < 0 then

• if D = 0 then

• if D > 0 then


(iv) Factoring polynomials


In practise: For quadratic polynomials f (x) = ax2 + bx + c

Calculate D = b2− 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


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 .


(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)(x2 + 1).

Example: f(x) = (x − 2)(x + 1)(x2− 3).

Check your understanding of Lecture 2

• The order of a polynomial:
What is the order of the following polynomials: f(x) = x4, f(x) = 2, f(x) = (x − 2)7,
f(x) = (x + 2)(x2 + 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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