Mathematics Examples
of Polynomials and
Inequalities
Dr Viktor Fedun
Automatic Control and Systems
Engineering, C09
Based on lectures by Dr Anthony Rossiter
Examples taken from the :
“Engineering Mathematics through Applications”
Kuldeep Singh Published by: Palgrave MacMillan
and http://tutorial.math.lamar.edu/
Example 1
(Page 145 Example 11)
[Mechanics]
The displacement, x(t), of a particle is given
by:
x(t)= (t-3)2
(a) Sketch the graph of displacement versus
time
(b) At what time(s) is x(t)=0?
Example 1
Solution
Solution:
(a) It is the same graph as the quadratic graph
t2 but shifted to the right by 3 units.
(b) x(t)=0 when t=3
Example 1
x(t)
0
Solution
x(t)= (t)2
3
x(t)= (t-3)2
t
Example 2
(Page 113 Exercise2 (c) q2
[Fluid Mechanics]
The velocity, v, of a fluid through a pipe is given by:
v = x2 – 9
Sketch the graph of v against x.
Example 2 Solution
v(x)
v(x)= x2
v(x)= x2-9
-3
3
0
-9
x
Example 3
(Page 118 Exercise 2(d) q3)
[Electrical principles]
The voltage, V, of a circuit is defined as:
V
= t2 – 5t + 6
(t ≥ 0)
Sketch the graph of V against t, indicating the
minimum value of V
Example 3 Solution
In order to plot the graph, it helps to find the values of t for
which the graph cuts the t axis, and the value of V for
which the graph crosses the V axis.
For the t axis, factorising the polynomial function and then
setting equal to zero will tell us of those values where
the graph crosses the t axis (i.e. when v=0).
V
= t2 – 5t + 6 (t ≥ 0)
=(t-2)(t-3)
So either (t-2)=0 or (t-3)=0 giving the crossings at t=2 and
t=3
For the V axis, the graph crosses the V axis when t=0,
giving V(t=0)=6
Example 3 Solution
v(t)
v(t)= t2-5t+6
6
0
Note: t ≥ 0
2
3
t
The minimum value is here
Polynomials
Functions made up of positive integer
powers of a variable, for instance:
y  x2
2
g  p 2p 5
q  s  2s  3
3
2
v  w  2w  w
5
z  4 ( yes even this)
Degree of a polynomial
The degree is the highest non-zero power
Degree of 1
Degree of 2
Degree of 5
Degree of 3
Degree of 0
y  x2
2
g  p 2p 5
q  s 5  2s  3
3
2
v  w  2w  w
z  4 ( yes even this)
Typical names
Degree of 0
Degree of 1
Degree of 2
Degree of 3
Degree of 4
Etc.
constant
linear
quadratic
cubic
quartic
Multiplying polynomials
If you multiply a rth order by an mth order, the
result has order r+m.
( x  a)(cx  b)  cx 2  acx  bx  ab
(ax 2  bx  c)( dx 3  x  e)  adx 5  bdx 4  (cd  a) x 3  (ae  b) x 2  
( x  a)( x  a)  x 2  a 2
In general, you do not want to do this by
hand, but you must be able to!
If you are not sure about multiplying out
brackets, see me asap.
Factorising a polynomial
Discuss in groups and prepare some
examples to share with the class.
1. What is a factor?
2. What is a factor of a polynomial?
3. What is the root of a polynomial?
4. What is the relationship between a factor
and a root?
5. How many factors/roots are there?
Finding factors/roots
We factorise a polynomial be writing it as a
product of 1st and/or 2nd order polynomials.
x  3x  2  ( x  2)( x  1)
3
2
x  1  ( x  1)( x  x  1)
4
2
2
2
2 x  16 x  30  ( x  3)( 2 x  10)
2
Finding factors/roots
We factorise a polynomial be writing it as a
product of 1st and/or 2nd order polynomials.
x  3x  2  ( x  2)( x  1)
3
2
x  1  ( x  1)( x  x  1)
4
2
2
2
2 x  16 x  30  ( x  3)( 2 x  10)
2
Factors
Factors are numbers (expressions) you can multiply
together to get another number (expressions):
2nd order polynomials are needed when this can not easily
be expressed as the product of two 1st order polynomials.
Finding factors/roots
A roots is defined as the values of
independent variable such that the
function is zero. i.e.
‘a’ is a root of f(x) if f(a)=0.
f ( x)  ( x  1)( x  2); f (1)  0, f (2)  0
f ( x)  4 x 3  32;
f (2)  0
f ( x)  3x 2  9 x  6; f (1)  0, f (2)  0
Finding factors/roots
Find factors and roots is the same problem.
1. A factor (x-a) has a root at ‘a’.
2. If a polynomial has roots at 2,3,5, the
polynomial is given as
f ( x)  A( x  2)( x  3)( x  5)
3. `A` cannot be determined solely from the
roots.
To factorise, first find the roots.
Problem
Define polynomials with roots:
• -1, -2 , 3
• 4, 5,-6,-7
Find the roots of the following polynomials
f ( x)  ( x  3)( x  2)( x  1)
f ( z )  ( z  1)( z  9)
2
What about quadratic factors
What are the roots of
How many roots does
an nth order
polynomial have?
f ( s)  s  2
2
What about quadratic factors
What are the roots of
How many roots does
an nth order
polynomial have?
f ( s)  s  2
2
Always n, but some
are not real numbers.
Solving for the roots with a clue
Find the roots of
f ( w)  w  3w  3w  1
3
2
Solving for the roots with a clue
Find the roots of
f ( w)  w  3w  3w  1
3
2
By inspection, one can see that w=-1 is a
root.
Solving for the roots with a clue
Find the roots of
f ( w)  w  3w  3w  1
3
2
By inspection, one can see that w=-1 is a
root. Therefore extract this factor, i.e.
( w  1)( Aw2  Bw  C )  w3  3w2  3w  1

Aw3  [ A  B]w2  [ B  C ]w  C  w3  3w2  3w  1
Hence, by inspection, A=1, B=2, C=1
Solving for the roots with a clue
Find the roots of
f ( w)  w  3w  3w  1
3
2
Given this quadratic factor, we can solve for
the remain two roots.
(w  1)( w  2w  1)  (w  1)
2
Hence, there are 3 roots at -1.
3
For the class
Solve for the roots of the following.
f ( x)  x  4 x  x  6
3
2
g ( p)  p  2 p  2 p  4
3
2
h( z )  3 z  6 z  2
2
Sketching polynomials
Sketch the following polynomials.
Key points to use are:
• Roots (intercept with horizontal axis).
• If order is even, increases to infinity for +ve and
–ve argument beyond domain of roots.
• If order is odd, one asymptote is + infinity and
the other is - infinity.
3
2
f ( x)  x  4 x  x  6
3
2
g ( p)  p  2 p  2 p  4
h( z )  3 z  6 z  2
2
Why are polynomials so important?
Within systems engineering, behaviour is
often reduced to solving for the roots of a
polynomial. Roots at (-a,-b) imply
behaviour of the form:
x(t )  Ae
 at
 Be
 bt
You must design the polynomial to have the
correct roots and hence to get the desired
behaviour from a system.
Inequalities
We will deal with
equations that involve
the symbols.
A key skill will be the
rearrangement of
functions.

  
What do these symbols
mean?
Discuss in class for 2
minutes.
Which of the following are true?
86
 8  6
( x  1)  (4 x  3)  2
( 2 x  4  2 )  x  1
Changing the order
In the following replace > by < or vice versa.
x  3  ??  ??
(2 x  1)  2  ??  ??
x  3  1  2 x  ??  ??
Linear Inequalities
Linear Inequalities
Linear Inequalities
Linear Inequalities
Linear Inequalities
Linear Inequalities
Example
Linear Inequalities
Example
Linear Inequalities
Example
or
Polynomial Inequalities
Example
Polynomial Inequalities
Example
Recipe
1. Get a zero on one side of the inequality
Polynomial Inequalities
Example
Recipe
1. Get a zero on one side of the inequality
2. If possible, factor the polynomial
Linear Inequalities
Example
Polynomial Inequalities
Example
Recipe
1. Get a zero on one side of the inequality
2. If possible, factor the polynomial
3. Determine where the polynomial is zero
Polynomial Inequalities
Example
Recipe
1. Get a zero on one side of the inequality
2. If possible, factor the polynomial
3. Determine where the polynomial is zero
4. Graph the points where the polynomial is zero
Polynomial Inequalities
Example
Recipe
4. Graph the points where the polynomial is zero
Polynomial Inequalities
Example
Recipe
4. Graph the points where the polynomial is zero
Polynomial Inequalities
For the class
Rational Inequalities
Rational Inequalities
Rational Inequalities
Rational Inequalities
Rational Inequalities
For the class
Rational Inequalities
For the class
Rational Inequalities
For the class
Rational Inequalities
For the class
Rational Inequalities
For the class
Absolute Value Equations
Absolute Value Equations
Absolute Value Equations
Absolute Value Equations
Absolute Value Equations
Absolute Value Inequalities
Absolute Value Inequalities
Absolute Value Inequalities
Absolute Value Inequalities
Absolute Value Inequalities
Absolute Value Inequalities
Absolute Value Inequalities
Absolute Value Inequalities