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Mathematics Examplesof Polynomials and
Inequalities
Examples taken from the :“Engineering Mathematics through Applications”
Kuldeep Singh Published by: Palgrave MacMillanand http://tutorial.math.lamar.edu/
Dr Viktor FedunAutomatic Control and Systems Engineering, C09
Based on lectures by Dr Anthony Rossiter
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 Solution
30
x(t)= (t-3)2x(t)= (t)2
t
x(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
3
-9
v(x)= x2-9
v(x)= x2
x
v(x)
-30
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
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
Example 3 Solution
3
6
v(t)= t2-5t+6
t
v(t)
20
The minimum value is hereNote: t ≥ 0
Polynomials
Functions made up of positive integer powers of a variable, for instance:
)(4
232
52
2
23
5
2
thisevenyesz
wwwvssq
ppg
xy
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 )(4
232
52
2
23
5
2
thisevenyesz
wwwvssq
ppg
xy
Typical names
Degree of 0constant
Degree of 1 linear
Degree of 2quadratic
Degree of 3cubic
Degree of 4quartic
Etc.
Multiplying polynomials
If you multiply a rth order by an mth order, the result has order r+m.
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.
22
234532
2
))((
)()())((
))((
axaxax
xbaexacdbdxadxexdxcbxax
abbxacxcxbcxax
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.
)102)(3(30162
)1)(1(1
)1)(2(23
2224
23
2
xxxx
xxxx
xxxx
Finding factors/roots
We factorise a polynomial be writing it as a product of 1st and/or 2nd order polynomials.
)102)(3(30162
)1)(1(1
)1)(2(23
2224
23
2
xxxx
xxxx
xxxx
Factors
2nd order polynomials are needed when this can not easily be expressed as the product of two 1st order polynomials.
Factors are numbers (expressions) you can multiply together to get another number (expressions):
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.
0)2(,0)1(;693)(
0)2(;324)(
0)2(,0)1();2)(1()(
2
3
ffxxxf
fxxf
ffxxxf
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
3. `A` cannot be determined solely from the roots.
)5)(3)(2()( xxxAxf
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
)9)(1()(
)1)(2)(3()(2
zzzf
xxxxf
What about quadratic factors
What are the roots of
How many roots does an nth order polynomial have?
2)( 2 ssf
What about quadratic factors
What are the roots of
How many roots does an nth order polynomial have?
2)( 2 ssf
Always n, but some are not real numbers.
Solving for the roots with a clue
Find the roots of 133)( 23 wwwwf
Solving for the roots with a clue
Find the roots of
By inspection, one can see that w=-1 is a root.
133)( 23 wwwwf
Solving for the roots with a clue
Find the roots of
By inspection, one can see that w=-1 is a root. Therefore extract this factor, i.e.
Hence, by inspection, A=1, B=2, C=1
133)( 23 wwwwf
133][][
133))(1(
2323
232
wwwCwCBwBAAw
wwwCBwAww
Solving for the roots with a clue
Find the roots of
Given this quadratic factor, we can solve for the remain two roots.
Hence, there are 3 roots at -1.
133)( 23 wwwwf
32 )1()12)(1( wwww
For the class
Solve for the roots of the following.
263)(
422)(64)(
2
23
23
zzzh
ppppgxxxxf
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.
263)(
422)(64)(
2
23
23
zzzh
ppppgxxxxf
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:
You must design the polynomial to have the correct roots and hence to get the desired behaviour from a system.
btat BeAetx )(
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?
1)242(
2)34()1(
68
68
xx
xx
Changing the order
In the following replace > by < or vice versa.
????213
????2)12(
????3
xx
x
x
Linear Inequalities
Linear Inequalities
Linear Inequalities
Linear Inequalities
Linear Inequalities
Linear InequalitiesExample
Linear InequalitiesExample
Linear InequalitiesExample
or
Polynomial InequalitiesExample
Polynomial InequalitiesExample
1. Get a zero on one side of the inequality
Recipe
Polynomial InequalitiesExample
1. Get a zero on one side of the inequality
Recipe
2. If possible, factor the polynomial
Linear InequalitiesExample
Polynomial InequalitiesExample
1. Get a zero on one side of the inequality
Recipe
2. If possible, factor the polynomial
3. Determine where the polynomial is zero
Polynomial InequalitiesExample
1. Get a zero on one side of the inequality
Recipe
2. If possible, factor the polynomial
3. Determine where the polynomial is zero
4. Graph the points where the polynomial is zero
Polynomial InequalitiesExample
Recipe
4. Graph the points where the polynomial is zero
Polynomial InequalitiesExample
Recipe
4. Graph the points where the polynomial is zero
Polynomial InequalitiesFor the class
Rational Inequalities
Rational Inequalities
Rational Inequalities
Rational Inequalities
Rational InequalitiesFor the class
Rational InequalitiesFor the class
Rational InequalitiesFor the class
Rational InequalitiesFor the class
Rational InequalitiesFor 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