The Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+

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1 The Cubic Formula The quadratic formula tells us the roots of a quadratic olynomial, a olynomial of the form ax + bx + c. The roots (if b b+ 4ac 0) are b 4ac a and b b 4ac a. The cubic formula tells us the roots of a cubic olynomial, a olynomial of the form ax + bx + cx + d. It was the invention (or discovery, deending on your oint of view) of the comlex numbers in the 16th century that allowed mathematicians to derive the cubic formula, and it was for this reason that eole became interested in comlex numbers. In this chater we ll see what the cubic formula is. We ll see how it uses comlex numbers to tell us the real number roots of cubic olynomials with real number coe cients. Secifically, we ll look at the two cubic olynomials x 15x 4andx x +, and we ll use comlex numbers and the cubic formula to determine what their real number roots are. A Comutation we ll need later Before we begin, we should make a quick digression. We ll want to find ( + i). We have two otions. We can multily it out as ( + i)( + i)( + i) using the distributive law many times. Or, we can use the binomial theorem. You may have seen the binomial theorem in Math It says that Simlifying, we have ( + i) +( )i +()i + i ( + i) 8+1i +6i + i 8+i1 + i 6+i i 8+i1 6 i +i11 We ll need to know that ( + i) +i11 later in this chater. 87

2 A Simlification for cubics The cubic formula tells us the roots of olynomials of the form ax + bx + cx + d. Equivalently, the cubic formula tells us the solutions of equations of the form ax + bx + cx + d 0. In the chater Classification of Conics, we saw that any quadratic equation in two variables can be modified to one of a few easy equations to understand. In a similar rocess, mathematicians had known that any cubic equation in one variable an equation of the form ax + bx + cx + d 0 could be modified to look like a cubic equation of the form x + ax + b 0, so it was these simler cubic equations that they were looking for solutions of. If they could find the solutions of equations of the form x + ax + b 0 then they would be able to find the solutions of any cubic equation. Of the simler cubic equations that they were trying to solve, there was an easier sort of equation to solve, and a more comlicated sort. The easier sort were equations of the form x + ax + b 0where a b ale 0. The a b more comlicated sort were equations x + ax + b 0where was a ositive number. We re going to focus on the more comlicated sort when discussing the cubic formula. Figuring out how to solve these more comlicated equations was the key to figuring out how to solve any cubic equation, and it was the cubic formula for these equations that lead to the discovery of comlex numbers. * * * * * * * * * * * * * 88

3 The Cubic formula Here are the stes for finding the roots of a cubic olynomial of the form x + ax + b if a b > 0 Ste 1. Let D be the comlex number D b + i r a b Ste. Find a comlex number z C such that z D. Ste. Let R be the real art of z, and let I be the imaginary art of z, so that R and I are real numbers with z R + ii. Ste 4. The three roots of x + ax + b are the real numbers R, R + I, and R I. These four stes together are the cubic formula. It uses comlex numbers (D and z) tocreaterealnumbers(r, R + I, and R I) thatare roots of the cubic olynomial x + ax + b. * * * * * * * * * * * * * 89

4 We ll take a look at two examles of cubic olynomials, and we ll use the cubic formula to find their roots. First examle In this examle we ll use the cubic formula to find the roots of the olynomial x 15x 4 Notice that this is a cubic olynomial x + ax + b where a b 4. Thus, a b 15 4 ( 5) ( ) ( 15) (4) and Because 11 is a ositive number, we can find the roots of the cubic olynomial x 15x 4usingthe4stesoutlinedonthereviousage. Ste 1. We need to find the comlex number D. It s given by D b + i r a b 4 + i 11 +i11 Ste. We need to find a comlex number z such that z +i11. We saw earlier in this chater that ( + i) +i11, so we can choose z +i Ste. In this ste, we write down the real (R) andimaginary(i) arts of z +i. The real art is, and the imaginary art is 1. So R and I 1 90

5 Ste 4. The three roots of the cubic olynomial x 15x 4arethe real numbers R ()4 R + I () + (1) + R I () (1) + We found the real number roots 4, +, and ofaolynomial that had real number coe cients 15 and 4 using comlex numbers + i11 and + i. This is why the comlex numbers seemed attractive to mathematicians originally. They cared about comlex numbers because they cared about real numbers, and comlex numbers were a tool designed to give them information about real numbers. Over the last few centuries, comlex numbers have roved their usefulness in mathematics in many other ways. They are now viewed as being just as imortant as the real numbers are. Second examle Next we ll find the roots of the cubic olynomial x x +. It s a olynomial of the form x + ax + b where a andb. To see if we can use the cubic formula on age 89, we need to see if the following number is ositive: 91

6 a b ( 1) ( ) ( 1) 1 1 Because 1 is ositive, we can roceed with the four stes of the cubic formula to find the roots of x x +. Ste 1. We need to find the number D. Using the equation from Ste 1 on age 89, D b + i r a r 1 + i 1 + i 1 + i 1 b Ste. We need to find a comlex number z such that z D. To solve this roblem, notice that D 1 + i 1 is the number in the unit 1 circle that is a counterclockwise rotation of 1 by the angle 4.From what we ve learned about multilying comlex numbers in the unit circle, we can see that we can choose z to be the number in the unit 9 4 oo

7 k circle obtained by rotating 1 by an angle of 1 we ll choose z to be the number 1 + i That is, It 0 It k It 0 It To reca, in Ste 1 we saw that D 1 + i 1. In Ste we saw that if z 1 + i 1, then z D. Ste. The real art of z 1 + i 1 is 1, and it s imaginary art also equals 1. That is, R I 1 Ste 4. The three real number roots of x 1 R x + are R + I R I

8 Exercises The first examle from this chater showed us the roots of x 15x 4. They are 4, +, and. Another way to ut this is to say that 4, +, and are the solutions of the equation x 15x 40. Here x is a variable, so we could equivalently write that 4, +, and arethesolutionsoftheequationz 15z 4 0, for examle. Use this to find the solutions of the following equations. For #1, first solve for x, then solve for x. For #, first solve for log e (x), and then solve for x. 1.) (x ) 15(x ) 40.) log e (x) 15 log e (x) 40 All further exercises in this chater have nothing to do with comlex numbers. 94

9 I Match the functions with their grahs..) f(x)+1 4.) f(x) 5.) f(x +1) I 6.) 1 f(x) 7.) f(x) 1 8.) f(x) 9.) f(x 1) 10.) f x f(x) A.) B.) a- J. C.) D.) E.) 74 a F.) G.) H.) N -a 1 a -. 95

10 Match the functions with their grahs. 11.) tan(x) 1.) cot(x) 1.) f(x) ( tan(x) if x ale 0; cot(x) if x>0. 14.) g(x) ( cot(x) if x<0; tan(x) if x 0. A.) B.) 7/7 /, C.) I Il I I! I Il II II ê I ILI I I 1 D.) 96

11 To find the solutions of an equation of the form h(x)f(x) h(x)g(x), you need to find the solutions of the following two equations: h(x) 0 and f(x) g(x) Find the solutions of the following equations. 15.) (x 1)(x +)(x 1)(x +4) 16.) (x )(x +1)(x )(x +) 17.) (x 7) (x 7)(x +) 18.) x x In the exercises from the revious chater we reviewed rules for when some equations have solutions that can be found in one ste. Those rules can be exanded on to give us rules for one art of a larger roblem that might involve several stes. These rules are listed below. Use them to solve the equations in the remaining exercises. f(x) c imlies f(x) c or f(x) c af(x) + bf(x)+c 0imliesf(x) b b 4ac a or f(x) b+ b 4ac a f(x) c imlies f(x) c 19.) (x ) 4.) ( x) 5 x ) x +4.) log e (x) 5 1.) (e x ) e x +0 4.) x 5 97