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Mathematics Examples of Polynomials and Inequalities Examples taken from the : “Engineering Mathematics through Applications” Kuldeep Singh Published by: Palgrave MacMillan and Dr Viktor Fedun Automatic Control and Systems Engineering, C09 Based on lectures by Dr Anthony Rossiter

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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 (a)Sketch the graph of displacement versus time (b)At what time(s) is x(t)=0?

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Example 1 Solution Solution: (a)It is the same graph as the quadratic graph t 2 but shifted to the right by 3 units. (b)x(t)=0 when t=3

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Example 1 Solution 3 0 x(t)= (t-3) 2 x(t)= (t) 2 t x(t)

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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 = x 2 – 9 Sketch the graph of v against x.

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Example 2 Solution 3 -9 v(x)= x 2 -9 v(x)= x 2 x v(x) -3 0

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Example 3 (Page 118 Exercise 2(d) q3) [Electrical principles] The voltage, V, of a circuit is defined as: V = t 2 – 5t + 6(t ≥ 0) Sketch the graph of V against t, indicating the minimum value of V

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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 = t 2 – 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

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3 6 v(t)= t 2 -5t+6 t v(t) 20 The minimum value is here Note: t ≥ 0

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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 Degree of 1 Degree of 2 Degree of 5 Degree of 3 Degree of 0

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Typical names Degree of 0constant Degree of 1linear Degree of 2quadratic Degree of 3cubic Degree of 4quartic Etc.

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Multiplying polynomials If you multiply a r th order by an m th 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. 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?

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

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

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

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

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Linear Inequalities Example

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Polynomial Inequalities Example 1. Get a zero on one side of the inequality Recipe 2. If possible, factor the polynomial 3. Determine where the polynomial is zero

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Polynomial Inequalities Example 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

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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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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 Inequalities

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