Calculus with Algebra and Trigonometry II
Calculus with Algebra and Trigonometry II MATH 217
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Date Created: 09/17/15
ALEC JOHNSON MATH 217 1 MAY 2007 WISCONSIN EMERGING SCHOLARS WORKSHEET 25 Review division factoring and root nding This is a brush up from last semester The division algorithm allows us to write any two polynomials x and px 7 0 uniquely in the form x px xx7 where qx is called the quo tient and xx is called the remainder In lecture we used the division algorithm to show that factoring is the same as root nding In particular7 the remainder theorem tells you that c is the remainder when you divide x by x 7 c To see this write x 7 6 qx x Then x Problems 1 Constructing a polynomial roots with given Construct a polynomial with real coef cients that has exactly the given zeros and degree a 3 7 42 degree 2 b 3 7 42 77 degree 3 c 3 7 42 77 degree 4 to Rational roots Completely factor the following polynomials HINT The rational roots theorem and synthetic division can be a big help a x37x24x74 b 2x3 7 5x2 711x 7 4 03 Complex arithmetic Let 21 2cis 150 and 22 3cis 760 Find a 21 and 22 in a bi form 2221 2221 ET O 25 CL e All solutions 2 to 22 21 F Derivative of a polynomial with a root of multiplicity greater than 1 Have each person in your group invent a poly nomial p that has a root of multiplicity 2 Now differentiate it to get the polynomial p Factor 19 Do you notice a common pattern Do you think that this happens in general Can you prove it 01 a 1 00 Factoring cubics part II Factoring a quadratic by depressing it Suppose you want to factor a quadratic polyno mial7 ie a polynomial of the form Ax2 Bx C If you divide by A7 you will get an equation of the form p x2 I x I 39y We wish to put this equation in the depressed form y2 c To ac complish this substitute x y 7 Oz and expand out to get an equation of the form y2 by c What should 04 be in order to make the y term zero Choose this a solve for x7 and expand your solution in terms of the original parameters of the quadratic polynomial AB7 and C Do you get a familiar formula Factoring cubics part I depressing a cubic Ever wonder how to get a general formula for the factors of a cubic polynomial Begin with a gen eral cubic7 Ax3 sz Ox 1 D Divide by A to get a cubic of the form x3 3x2 39yx 6 We wish to put this equation in the depressed form y3 I 0y I d The trick is to make a substitution of the form x y 7 04 What should 04 be in order to make the y2 term zero solving a de pressed cubic Suppose you are trying to factor a depressed cu bic7 p y3 I 0y I 1 So you are trying to solve the equation y3 I 0y I d O The trick is to look for a solution of the form y s 7 t Plug this in Show that we ve got a solution if 32f37337 d3st So if we can solve this system for s and t in terms of c and d7 we re done Do you see how to solve it Factoring cubics part III try an example Now let s do an example To have a nice example to test out our method7 let s make up a cubic where we already know the roots I don t know7 how about px x 7 3x2 1 Or maybe you would prefer real roots7 eg px x 7 5x 1x 4 Multiply out the factored polynomial that you pick and then use the methods in the previous two sections to factor it Does it work You also might try starting with a random de pressed cubic7 such as px x3 7 3x 1 Com pute the roots Then nd the zeros by graphing or using Newton s method Do your answers agree Factoring a quartic O To factor a quartic equation rst gure out a substitution that depresses it ie gets rid of the cube term Then assume that you can write it as a product of quadratic polynomials with generic coe icients Multiply the quadratics to get a quartic and equate the coef cients with the coef cients of the depressed quartic This gives a system of equations which with some strate gic elimination of variables you can reduce to the problem of nding the roots of a cubic Voila Factoring a quintic Show that it is impossible to represent the roots of the polynomial x 5 7 z 7 1 using addi tion subtraction multiplication diVision or ex traction of roots Note that this result shows that it is impossible to nd a general formula for the roots of a polynomial of degree 5 or higher Hint read up on Galois theoryl