Presentation on theme: "Mathematics Examples of Polynomials and Inequalities"— Presentation transcript:
1 Mathematics Examples of Polynomials and Inequalities Dr Viktor FedunAutomatic Control and Systems Engineering, C09Based on lectures by Dr Anthony RossiterExamples taken from the :“Engineering Mathematics through Applications”Kuldeep Singh Published by: Palgrave MacMillanand
2 Example 1 (Page 145 Example 11) [Mechanics]The displacement, x(t), of a particle is given by:x(t)= (t-3)2Sketch the graph of displacement versus timeAt what time(s) is x(t)=0?
3 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
7 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
8 Example 3 SolutionIn 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=3For the V axis, the graph crosses the V axis when t=0, giving V(t=0)=6
9 Example 3 Solution v(t) v(t)= t2-5t+6 6 2 3 t Note: t ≥ 0 23tNote: t ≥ 0The minimum value is here
10 PolynomialsFunctions made up of positive integer powers of a variable, for instance:
11 Degree of a polynomial The degree is the highest non-zero power
12 Typical names Degree of 0 constant Degree of 1 linear Degree of 2 quadraticDegree of 3 cubicDegree of 4 quarticEtc.
13 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.
14 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?
15 Finding factors/roots We factorise a polynomial be writing it as a product of 1st and/or 2nd order polynomials.
16 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):Factors2nd order polynomials are needed when this can not easily be expressed as the product of two 1st order polynomials.
17 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.
18 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.
19 Problem Define polynomials with roots: -1, -2 , 3 4, 5,-6,-7 Find the roots of the following polynomials
20 What about quadratic factors What are the roots ofHow many roots does an nth order polynomial have?
21 What about quadratic factors What are the roots ofHow many roots does an nth order polynomial have?Always n, but some are not real numbers.
22 Solving for the roots with a clue Find the roots of
23 Solving for the roots with a clue Find the roots ofBy inspection, one can see that w=-1 is a root.
24 Solving for the roots with a clue Find the roots ofBy 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
25 Solving for the roots with a clue Find the roots ofGiven this quadratic factor, we can solve for the remain two roots.Hence, there are 3 roots at -1.
26 For the classSolve for the roots of the following.
27 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.
28 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.
29 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.
44 Polynomial Inequalities Example Recipe1. Get a zero on one side of the inequality2. If possible, factor the polynomial3. Determine where the polynomial is zero
45 Polynomial Inequalities Example Recipe1. Get a zero on one side of the inequality2. If possible, factor the polynomial3. Determine where the polynomial is zero4. Graph the points where the polynomial is zero
46 Polynomial Inequalities Example Recipe4. Graph the points where the polynomial is zero
47 Polynomial Inequalities Example Recipe4. Graph the points where the polynomial is zero