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**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

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**Example 1 (Page 145 Example 11)**

[Mechanics] The displacement, x(t), of a particle is given by: x(t)= (t-3)2 Sketch the graph of displacement versus time At what time(s) is x(t)=0?

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**Example 1 Solution Solution:**

It is the same graph as the quadratic graph t2 but shifted to the right by 3 units. x(t)=0 when t=3

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Example 1 Solution x(t) x(t)= (t)2 x(t)= (t-3)2 t 3

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**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.

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Example 2 Solution v(x) v(x)= x2 v(x)= x2-9 -3 3 x -9

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**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

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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

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**Example 3 Solution v(t) v(t)= t2-5t+6 6 2 3 t Note: t ≥ 0**

2 3 t Note: t ≥ 0 The minimum value is here

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Polynomials Functions made up of positive integer powers of a variable, for instance:

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**Degree of a polynomial The degree is the highest non-zero power**

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**Typical names Degree of 0 constant Degree of 1 linear**

Degree of 2 quadratic Degree of 3 cubic Degree of 4 quartic Etc.

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**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.

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**Factorising a polynomial**

Discuss in groups and prepare some examples to share with the class. What is a factor? What is a factor of a polynomial? What is the root of a polynomial? What is the relationship between a factor and a root? How many factors/roots are there?

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**Finding factors/roots**

We factorise a polynomial be writing it as a product of 1st and/or 2nd order polynomials.

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**Finding factors/roots**

We factorise a polynomial be writing it as a product of 1st and/or 2nd order polynomials. Factors are numbers (expressions) you can multiply together to get another number (expressions): Factors 2nd order polynomials are needed when this can not easily be expressed as the product of two 1st order polynomials.

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**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.

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**Finding factors/roots**

Find factors and roots is the same problem. A factor (x-a) has a root at ‘a’. If a polynomial has roots at 2,3,5, the polynomial is given as `A` cannot be determined solely from the roots. To factorise, first find the roots.

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**Problem Define polynomials with roots: -1, -2 , 3 4, 5,-6,-7**

Find the roots of the following polynomials

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**What about quadratic factors**

What are the roots of How many roots does an nth order polynomial have?

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**What about quadratic factors**

What are the roots of How many roots does an nth order polynomial have? Always n, but some are not real numbers.

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**Solving for the roots with a clue**

Find the roots of

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**Solving for the roots with a clue**

Find the roots of By inspection, one can see that w=-1 is a root.

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**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

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**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.

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For the class Solve for the roots of the following.

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**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.

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**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.

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**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.

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**Which of the following are true?**

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Changing the order In the following replace > by < or vice versa.

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Linear Inequalities

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Linear Inequalities

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Linear Inequalities

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Linear Inequalities

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Linear Inequalities

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Linear Inequalities Example

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Linear Inequalities Example

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Linear Inequalities Example or

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**Polynomial Inequalities**

Example

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**Polynomial Inequalities Example**

Recipe 1. Get a zero on one side of the inequality

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**Polynomial Inequalities Example**

Recipe 1. Get a zero on one side of the inequality 2. If possible, factor the polynomial

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Linear Inequalities Example

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**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

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**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

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**Polynomial Inequalities Example**

Recipe 4. Graph the points where the polynomial is zero

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**Polynomial Inequalities Example**

Recipe 4. Graph the points where the polynomial is zero

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**Polynomial Inequalities**

For the class

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**Rational Inequalities**

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**Rational Inequalities**

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**Rational Inequalities**

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**Rational Inequalities**

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**Rational Inequalities**

For the class

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**Rational Inequalities**

For the class

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**Rational Inequalities**

For the class

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**Rational Inequalities**

For the class

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**Rational Inequalities**

For the class

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**Absolute Value Equations**

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**Absolute Value Equations**

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**Absolute Value Equations**

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**Absolute Value Equations**

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**Absolute Value Equations**

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**Absolute Value Inequalities**

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**Absolute Value Inequalities**

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**Absolute Value Inequalities**

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**Absolute Value Inequalities**

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**Absolute Value Inequalities**

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**Absolute Value Inequalities**

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**Absolute Value Inequalities**

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**Absolute Value Inequalities**

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