mathematics examples of polynomials and inequalities

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