Mths formul book Nick Urbnik In memory of Gunter Beck, gret techer
1 Mths formul book Contents 1 Algebric Results 3 Absolute vlue 3 3 Inequlities 4 4 Trigonometry 4 4.1 Sum n prouct formuls..................... 8 4. Sum of two wves.......................... 9 5 Coorinte geometry: stright lines 10 6 Logrithms n Inexes 11 7 Series n Sequences 1 8 Finnce 14 9 Qurtic equtions 15 10 Differentition 15 10.1 Derivtives of trig functions.................... 16 10. Derivtives of exponentil n log functions........... 17 10.3 Sttionry points.......................... 17 11 Integrtion 18 11.1 Integrtion by prts........................ 18 11. Numericl integrtion....................... 18 11.3 Inefinite integrls......................... 18 11.4 Definite integrls.......................... 0 1 Ares n volumes of geometric shpes 0 1.1 Volumes............................... 1 13 Simple hrmonic motion 1 14 Newton s metho 15 Binomil 15.1 Binomil theorem.......................... 16 Hyperbolic functions 3
Mths formul book 17 Sets of numbers 3 18 Complex numbers 3 19 Polynomils 5 19.1 Multiple roots & erive polynomils............... 6 19. Prtil frctions........................... 7 Introuction This is collection of formuls given by my mthemtics techer, Gunter Beck, t Meowbnk Technicl College, in Austrli, uring his teching of four unit mths (Higher School Certificte). As he introuce ech topic, he wrote the min formuls for tht topic on the bor, n encourge us to write them into our own formul books. He ie rther quickly from cncer. He ws my fvourite techer. After I lerne of his eth, n notice tht my ol formul book ws getting very og ere from so mny yers of constnt use, I ecie to write this in fon memory of Gunter, using AMS-L A TEX. There re mny who miss you, Gunter.
3 Mths formul book 1 Algebric Results + c b Fctors of x n n if b = c then: = bc igonl prouct (1.1) c = b b = c = c + igonl exchnge (1.) inverse (1.3) en (1.4) x n n = (x )(x n 1 + x n + x n 3 +... + x n + n 1 ) (1.5) Sum n ifference of two cubes 3 + b 3 = ( + b)( b + b ) (1.6) 3 b 3 = ( b)( + b + b ) (1.7) b = ( + b)( b) (1.8) Expnsions of vrious squres n cubes ( + b + c + + n) = + b + c + + n + b (where b represents ll possible pirs of, b, c,..., n) (1.9) ( + b + c + ) = + b + c + + b + c + + bc + b + c (1.10) Absolute vlue ( + b + c) = + b + c + b + c + bc (1.11) ( b) 3 = 3 3 b + 3b b 3 (1.1) ( + b) 3 = 3 + 3 b + 3b + b 3 (1.13) Definition.1 The efinition of the bsolute vlue of is if > 0 = 0 if = 0 if < 0 (.1)
Mths formul book 4 Some properties if x = then x = ± (.) if x < then < x < (.3) if x > then x < or x > (.4) 3 Inequlities If > b, c >, then: ± c > b ± c (3.1) c > bc (c > 0) (3.) c < bc (c < 0) (3.3) c > b (, b, c, ll > 0) (3.4) > b (, b both > 0) (3.5) 1 < 1 b (, b both > 0) (3.6) 4 Trigonometry All Sttions To Centrl : S T A C Reciprocl rtios csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tn θ = cos θ sin θ tn θ = sin θ cos θ (4.1) (4.) (4.3) (4.4)
5 Mths formul book Pythgoren ientities sin θ + cos θ = 1 (4.5) tn θ + 1 = sec θ (4.6) 1 + cot θ = csc θ (4.7) Inverse formuls sin 1 x = tn 1 x 1 x cos 1 x = tn 1 1 x x x 1 (4.8) x 0 (4.9) Co-rtios (complementry ngles) sin(90 θ) = cos θ cos(90 θ) = sin θ (4.10) tn(90 θ) = cot θ cot(90 θ) = tn θ (4.11) sec(90 θ) = csc θ csc(90 θ) = sec θ (4.1) sin(180 θ) = sin θ sin(180 + θ) = sin θ (4.13) cos(180 θ) = cos θ cos(180 + θ) = cos θ (4.14) tn(180 θ) = tn θ tn(180 + θ) = tn θ (4.15) sin(360 θ) = sin θ = sin θ (o) (4.16) cos(360 θ) = cos θ = cos θ (even) (4.17) tn(360 θ) = tn θ = tn θ (o) (4.18) Exct vlues 1 45 1 1 60 3 30
Mths formul book 6 sin 45 = cos 45 = 1 (4.19) sin 30 = cos 60 = 1 (4.0) 3 sin 60 = cos 30 = (4.1) tn 45 = cot 45 = 1 (4.) tn 30 = cot 60 = 1 3 (4.3) tn 60 = cot 30 = 3 (4.4) 6 sin 15 = cos 75 = (4.5) 4 6 + sin 75 = cos 15 = (4.6) 4 tn 15 = cot 75 = 3 (4.7) tn 75 = cot 15 = + 3 (4.8) Aition formuls sin(θ ± φ) = sin θ cos φ ± cos θ sin φ (4.9) cos(θ ± φ) = cos θ cos φ sin θ sin φ (4.30) tn θ ± tn φ tn(θ ± φ) = 1 tn θ tn φ (4.31) cot θ cot φ 1 cot(θ ± φ) = sin θ cot θ ± cot φ = sin θ cos θ (4.3) cos θ = cos θ sin θ (4.33) = 1 sin θ (4.34) = cos θ 1 (4.35) tn θ = tn θ 1 tn θ sin 1 cos θ θ = cos 1 + cos θ θ = (4.36) (4.37) (4.38)
7 Mths formul book Sine rule sin A = b sin B = c sin C (4.39) Are of tringle Are of ny tringle = b sin C (4.40) Cosine rule = b + c bc cos A (sie formul) (4.41) cos A = b + c bc (ngle formul) (4.4) Little t formul 1 + t t x 1 t t = tn x (4.43) if t = tn x tn x = tn x 1 tn x (4.44) = t 1 t (4.45) sin x = t 1 + t (4.46) cos x = 1 t 1 + t (4.47) then x = t 1 + t (4.48)
Mths formul book 8 Perio n mplitue y = sin(ωt + φ) y = cos(ωt + φ) y = tn(ωt + φ) mplitue =, perio = π ω mplitue =, perio = π ω mplitue =, perio = π ω (4.49) (4.50) (4.51) Specil limits sin θ lim θ 0 θ lim θ 0 tn θ θ = 1 (4.5) = 1 (4.53) The 3θ results sin 3θ = 3 sin θ 4 sin 3 θ (4.54) cos 3θ = 4 cos 3 θ 3 cos θ (4.55) tn 3θ = 3 tn θ tn3 θ 1 3 tn θ 4.1 Sum n prouct formuls (4.56) if { 1 = θ + φ = θ φ sin(θ + φ) + sin(θ φ) = sin θ cos φ (4.57) sin(θ + φ) sin(θ φ) = sin φ cos θ (4.58) cos(θ + φ) + cos(θ φ) = cos θ cos φ (4.59) cos(θ + φ) cos(θ φ) = sin θ sin φ (4.60) then θ = 1 ( 1 + ) φ = 1 ( 1 ) use θ > φ (4.61) Using these formuls: sin = ei e i i cos = ei + e i (4.6) (4.63)
9 Mths formul book it is not too hr to erive the sum n prouct formuls, e.g., ( cos cos b = 1 4 e i + e i) ( e ib + e ib) (e i(+b) + e i( b) + e i( b) + e i(+b)) = 1 4 = 1 4 ( e i(+b) + e i(+b)) + 1 4 (e i( b) + e i( b)) = 1 cos( + b) + 1 cos( b). 4. Sum of two wves cos ωt + b sin ωt = + b cos ( ωt tn 1 b ) (4.64) (4.65) Here we use the fct tht n + jb = + b e j tn 1 b (4.66) jb = + jb = + b e j tn 1 b (4.67) (4.68) cos ωt + b sin ωt = ( e jωt + e jωt) + b ( e jωt e jωt) (4.69) j = ( e jωt + e jωt) jb ( e jωt e jωt) (4.70) = 1 ( e jωt ( jb) + e jωt ( + jb) ) (from (4.66) n (4.67)) (4.71) = 1 ( ) + b e jωt e j tn 1 b + e jωt e j tn 1 b (4.7) + b = ( e j(ωt tn 1 b ) + e j(ωt tn 1 b )) (4.73) = ( + b cos ωt tn 1 b ) (4.74) Perio of sum of two sine wves: If cos ω 1 t + cos t is perioic with perio T, then m, n Z : ω 1 T = πm, ω T = πn, i.e., ω 1 = m is rtionl. ω n Also, to fin T : ω = n m ω 1, T = π m = π n. ω 1 ω
Mths formul book 10 The sme is true of cos(ω 1 t+θ), cos(ω t+φ), ffect the vlue of T here. θ, φ R, i.e., phse oesn t 5 Coorinte geometry: stright lines Equtions of stright lines The eqution of line y xis through (, b) is y = b (5.1) The eqution of line x xis through (, b) is The eqution of line with slope m n y intercept b is x = (5.) y = mx + b (5.3) The eqution of line with slope m through (x 1, y 1 ) is y y 1 = m(x x 1 ) (5.4) The eqution of line through (x 1, y 1 ) n (x, y ) is y y 1 = y y 1 x x 1 (x x 1 ) (5.5) The eqution with x intercept n y intercept b is x + y b = 1 (5.6) This generl form of eqution for line hs slope = A C B, x intercept = A, y intercept = C B : Ax + By + C = 0 (5.7) Distnce n point formuls The istnce between (x 1, x ) n (x, y ) is (x x 1 ) + (y y 1 ) (5.8) The istnce of (x 1, y 1 ) from Ax + By + C = 0 is = Ax 1 + By 1 + C (5.9) A + B
11 Mths formul book The coorintes of the point which ivies the join of Q = (x 1, y 1 ) n R = (x, y ) in the rtio m i : m is ( m1 x + m x 1 P =, m ) 1y + m y 1 (5.10) m 1 + m m 1 + m From Q to P to R. If P is outsie of the line QR, then QP, RP re mesure in opposite senses, n m1 m is negtive. The coorintes of the mipoint of the join of (x 1, y 1 ) n (x, y ) is ( x1 + x, y ) 1 + y (5.11) The ngle α between two lines with slopes m 1, m is given by tn α = m m 1 1 + m m 1 for α cute (5.1) tn α = m m 1 1 + m m 1 for α obtuse (5.13) 6 Logrithms n Inexes Bsic inex lws m n = m+n (6.1) m n = m n (6.) ( m ) n = mn (6.3) 0 = 1 (6.4) n = 1 n (6.5) m n = n m (if m > 0) (6.6) (b) n = n b n (6.7) n n b = n (6.8) b Definition 6.1 (Definition of logrithm) if log x = y then x = y. The number is clle the bse of the logrithm. Chnge of bse of logrithms log x = log b x log b (6.9)
Mths formul book 1 Bsic lws of logrithms log mn = log m + log n (6.10) m log n = log m log n (6.11) log m x = x log m (6.1) 7 Series n Sequences x = e x ln (6.13) x e = e e log x (6.14) e x = 1 + x + x! + x3 + x. (6.15) 3! Arithmetic progression (AP): is of the form:, +, +,..., u n u n = + ( 1) (7.1) Tests for rithmetic progression Sum of n AP u u 1 = u 3 u = = u n u n 1 if, b, c re sequentil terms in AP, then b = + c S n = n ( ) + (k 1) k=1 (7.) u n = S n S n 1 (7.3) S n = n( + u n) S n = n( + (n 1) ) if given lst term u n (7.4) if lst term not given (7.5)
13 Mths formul book Geometric progression (GP): is of the form:, r, r,..., r n 1 u n = r n 1 (7.6) Test for GP u u 1 = u 3 u = = u n u n 1 u = u 1 u 3,..., u n 1 = u n u n Sum of GP S n = n k=1 r k 1 S n = (rn 1) r 1 S n = (1 rn ) 1 r use if r > 1 (7.7) use if r < 1 (7.8) Sum to infinity S = 1 r where r < 1. (7.9) n r k = r k=1 Some other useful sums n k = k=0 n i=0 n k=1 r n 1 = r(rn 1) r 1 = rn+1 r r 1 n(n + 1)(n + 1) 6 k 3 = n (n + 1) 4 n k = k=0 n(n + 1) (7.10) (7.11) (7.1) (7.13)
Mths formul book 14 e x = 1 + x + x! + x3 3! +... (7.14) x = e x ln (x ln ) (x ln )3 = 1 + x ln + + +...! 3! (7.15) ln(1 + x) = x x + x3 3 x4 +..., 1 < x 1. (7.16) 4 Series relevnt to Z trnsforms z (1 z) Now M z n = 1 + z + z + z 3 + + z M 1 + z M (7.17) n=0 M z n = z + z + z 3 + + z M 1 + z M+1 n=0 multiplying b.s. by z (7.18) M z n = 1 z M+1 subtrcting (7.17) (7.18) n=0 M n=0 z n = 1 zm+1 1 z z n = n=0 lim M n=0 (7.19) (7.0) M z n (7.1) 1 z M+1 = lim (7.) M 1 z = 1 z : z < 1 (7.3) 1 z This is from Thoms & Finney, pges 610 611. 8 Finnce Simple interest I = P tr (8.1) where P = principl, t = time, r = rte (i.e., 6% 6 100 )
15 Mths formul book Compoun interest A = P (1 + r) n (8.) where P = principl, n = number of time perios, r = rte per time perio, A = mount you get. Reucible interest: This uses S n = (rn 1) r 1 M ( (1 + R 100 )n 1 ) R 100 = P where is M, r = 1 + R 100. ( 1 + R ) n (8.3) 100 i.e., M = P (1 + R 100 )n R 100 (1 + R 100 )n 1 where M = equl instllment pi per unit time R = % rte of interest per unit time (Note: sme unit of time s for M) n = number of instllments. (8.4) Supernnution S n = ( ( A 1 + R 100 (1 + R 100 )n 1 )) R 100 (8.5) where A is n equl instllment per unit time. n, R re s bove. 9 Qurtic equtions The solution of x + bx + c = 0 is x = b ± b 4c (9.1) 10 Differentition From first principles: f f(x + h) f(x) (x) = lim h 0 h or f f(x) f(c) (c) = lim x c x c (10.1) (10.)
Mths formul book 16 Prouct rule x xn = nx n 1 (10.3) x c = 0 (10.4) x c f(x) = c f (x) (10.5) ( ) f(x) ± g(x) = f (x) ± g (x) x (10.6) v (uv) = u x x + v u x w v u (uvw) = uv + uw + vw x x x x (10.7) (10.8) Quotient rule u x v = v u x u v x v (10.9) Chin rule (Function of function rule, composite function rule, substitution rule) y x = y u u (10.10) x 10.1 Derivtives of trig functions sin x = cos x x (10.11) cos x = sin x x (10.1) x tn x = sec x (10.13) 1 x sin 1 x = 1 x π < sin 1 x < π (10.14) x cos 1 x = 1 1 x π < cos 1 x < π (10.15) x tn 1 x = 1 1 + x π < tn 1 x < π (10.16) sec x = sec x tn x x (10.17)
17 Mths formul book csc x = csc x cot x x (10.18) x cot x = csc x (10.19) sinh x = cosh x x (10.0) cosh x = sinh x x (10.1) 10. Derivtives of exponentil n log functions x ln f(x) = f (x) f(x) (10.) x ef(x) = f (x)e f(x) (10.3) x ex = e x (10.4) x x = ln x since k = e k ln (10.5) y ln y (10.6) x ln y = x x ln y = 1 y 10.3 Sttionry points y x x (10.7) f (x) = 0 t sttionry point of f(x). (10.8) For sttionry point (w, f(w)): if grient chnges sign roun (w, f(w)), locl mximum or minimum (10.9) if grient oes not chnge sign roun (w, f(w)), point of inflection. (10.30) if f (w) > 0 then (w, f(w)) is minimum point. (10.31) if f (w) < 0 then (w, f(w)) is mximum point. (10.3) if f (w) = 0 then it coul be minimum, mximum or point of inflection. (10.33)
Mths formul book 18 11 Integrtion 11.1 Integrtion by prts u v x = uv x v u x (11.1) x 11. Numericl integrtion Trpezoil rule b Simpson s rule f(x) x b ( y0 + y n + (y 1 + y + y 3 + + y n 1 ) ) (11.) n b f(x) x b 3n (y 0+4y 1 +y +4y 3 +y 4 + +4y n 1 +y n ) n is even (11.3) 11.3 Inefinite integrls K x = Kx + c (11.4) x n x = xn+1 n + 1 + c (11.5) (x + b) n (x + b)n+1 x = + c (11.6) (n + 1) sin x x = 1 cos x + c (11.7) cos x x = 1 sin x + c (11.8) sec x x = 1 tn x + c (11.9) e x x = 1 ex + c (11.10) x = ln x + c (11.11) x f (x) x = ln f(x) + c (11.1) f(x)
19 Mths formul book x 1 x x = sin 1 x + c (11.13) 1 x = 1 x cos 1 x + c (11.14) x 1 + x = tn 1 x + c (11.15) x x = x sin 1 + c (11.16) = cos 1 x + c, < x < (11.17) x + x = 1 x tn 1 + c (11.18) e u u = e u + c, i.e., e f(x) f (x) x = e f(x) + c (11.19) e x x = ex + c (11.0) 1 x x = 1 log x + + c (11.1) x x = 1 log + x x + c (11.) ln x x = x ln x x + c (11.3) x + x = log(x + + x ) + c (11.4) x x = log(x + x ) + c x : x > (11.5) tn x x = log sec x + c (11.6) sec x x = ln(sec x + tn x) + c (11.7) = ln(sec x tn x) + c (11.8) ( π = ln tn 4 + x ) + c (11.9) cot x x = ln sin x + c (11.30) csc (x) x = 1 cot(x) + c (11.31)
Mths formul book 0 x x = e x ln x = ex ln ln + c = x ln + c (11.3) (let y = x, then ln y = x ln, so y = e x ln ). (11.33) cosh x x = 1 sinh x + c (11.34) sinh x x = 1 cosh x + c (11.35) Trig substitutions for + x try x = tn θ (11.36) x try x = sin θ (11.37) x + try x = sec θ (11.38) if t = tn x then x = t 1 + t (11.39) 11.4 Definite integrls { 0 x 0 } f(t) t = f(x) (11.40) f(x) x = 0 f( x) x (11.41) 1 Ares n volumes of geometric shpes Length of n rc with ngle θ n rius r is l = rθ. Are of sector ngle θ n rius r is A = 1 r θ Are of segment of circle with ngle θ n rius r is A = 1 r (θ sin θ) Are of tringle = 1 r sin θ re of segment = 1 r θ 1 r sin θ Are of tringle with height h n bse b is A = 1 hb Are of prllelogrm with height h n bse b is A = bh
1 Mths formul book Are of trpezium with prllel sies of length, b n height h is A = 1 h( + b) Are of rhombus with igonls of length, b is A = 1 b Are of circle is A = πr, circumference is C = πr Are of n ellipse with semi-mjor xis length, semi-minor xis length b is A = πb. (Note tht, b become the rius s the ellipse pproches the shpe of circle.) 1.1 Volumes In the following, h is the height of the shpe, r is rius. Right rectngulr prism or oblique rectngulr prism with bse lengths n b: V = bh Cyliner with circulr bse, both upright n oblique, V = πr h Pyrmis with rectngulr bses of sie length, b, both right pyrmis n oblique pyrmis: V = 1 3 bh Cones: V = 1 3 πr h The curve surfce re of n upright cone with length from pex to ege of the bse s (s is not the height, but the length of the sie of the cone): S = πrs. Volume of sphere V = 4 3 πr3 Surfce re of sphere is S = 4πr 13 Simple hrmonic motion Definition 13.1 ẍ = n x n is ny constnt (13.1) Other properties v = n ( x ) = mplitue (13.) x = cos nt (13.3) T = π n T is the perio (13.4) v mx = n (13.5) ẍ mx = n (13.6)
Mths formul book 14 Newton s metho If x 1 is goo pproximtion to f(x) = 0 then x = x 1 f(x 1) f (x 1 ) (14.1) is better pproximtion. 15 Binomil The coefficients in the expnsion of ( + b) n where n = 0, 1,, 3,... re given by Pscl s tringle. 15.1 Binomil theorem ( + b) n = = n T r+1 (15.1) r=0 n r=0 n! r!(n r)! n r b r (15.) T r+1 = n C r n r b r (15.3) ( ) n where n n! C r = = (15.4) r r!(n r)! Pscl s tringle reltionship n+1 C r = n C r 1 + n C r n (15.5) Sum of coefficients n n C r = n (15.6) r=0 Symmetricl reltionship n C r = n C n r (15.7)
3 Mths formul book 16 Hyperbolic functions Ientities cosh x = ex + e x even: ctenry (16.1) sinh x = ex e x o: 1 1, onto. (16.) cosh x = sinh x x (16.3) sinh x = cosh x x (16.4) cosh x sinh x = 1 (16.5) cosh x = cosh x + sinh x (16.6) sinh(x ± y) = sinh x cosh y ± cosh x sinh y (16.7) 17 Sets of numbers subset nottion elements Complex numbers C Rel numbers R Nturl numbers (or whole numbers) N 1,, 3, 4, 5,... Integers Z...,, 1, 0, 1,, 3,... Rtionl numbers (or frctions) Q 0, 1,, 1, 1, 3 4, 5 3, 1, 3 7,... Reference: K. G. Binmore, Mthemticl Anlysis: strightforwr pproch, Cmbrige, 1983. Note tht Gunter clle the set of integers J. He lso referre to set R, n clle the set of rtionls Q or R. He inclue 0 in the set of Crinls or nturl numbers, N. 18 Complex numbers Definition of i Definition 18.1 1 = ±i (18.1)
Mths formul book 4 Equlity Definition 18. If + ib = c + i, then = c n b =. Mo-rg theorems Theorem 18.1 z 1 z = z 1 z (18.) rg(z 1 z ) = rg z1 + rg z (18.3) Theorem 18. z 1 z = z 1 z (18.4) rg z 1 z = rg z1 rg z (18.5) Euler s formul cos θ + i sin θ = cis θ = e iθ (18.6) Gunter clle cos θ + i sin θ cis θ. e Moivre s theorem Theorem 18.3 cos nθ + in sin θ = cis nθ = e inθ (18.7) Cube roots of unity: if z 3 = 1 then z = 1 or 1 ± i 3. Ech root is the squre of the other. Using e Moivre s theorem: Here we use z 3 = 1 = z 3 = 1 = z 3 = 1 = z = 1.
5 Mths formul book let z = r 3 (cos θ + i sin θ) 3 further, z = cos 3θ + i sin 3θ cos 3θ + i sin 3θ = 1 + i0 Now cos 3θ = 1, i sin 3θ = i0 3θ = 0, π, 4π,... θ = 0, π 3, π 3 Hence z = cis 0, cis π 3 ( cis π ) = cis 4π 3 3 = cis π 3, cis π 3 one complex root is the squre of the other. Now r = 1, s bove (by e Moivre s Theorem) since θ < π. (e Moivre s Theorem) = other cube root of unity. This sme metho is use to fin the nth root of unity. See section 18 on the fcing pge for the mening of cis θ. Multiplying complex number by i rottes the number by π (nti-clockwise) sense: in positive z + z = R(z) (18.8) z z = ii(z) (18.9) z = z z (18.10) rg z 1 = rg z (18.11) iω = iω (18.1) 19 Polynomils Definition 19.1 Monic mens the leing coefficient n = 1. Theorem 19.1 Every polynomil over F cn be fctorise into monic fctors of egree 1 n constnt Reminer theorem: Theorem 19. A(x) is polynomil over F; A(x) = A() when A(x) is ivie by (x ).
Mths formul book 6 The fctor theorem Theorem 19.3 (x ) is fctor of A(x) iff A() = 0 Theorem of rtionl roots of polynomils Theorem 19.4 If r s is zero of P (x) = nx n + n 1 x n 1 + + 1 x + 0 ( n 0), r, s re reltively prime n 0, 1,..., n Z, then s/ n n r/ 0, i.e., (sx r) is fctor of P (x). Funmentl theorem of lgebr Theorem 19.5 Every P (x) with coefficients over R, R or C hs root (P (α) = 0) for some complex number α. See section 17 on pge 3 for the prticulr mening of R, R here. Theorem 19.6 A polynomil P (x) of egree n > 0 hs exctly n zeros in fiel C n hence exctly n liner fctors over C. Theorem 19.7 If α = + ib is root of P (x) (whose coefficients re rel), then its complex conjugte, α = ib is lso root. Theorem 19.8 If P (x) hs egree n with rel coefficients, if n is o there is lwys t lest one rel root. 19.1 Multiple roots & erive polynomils Theorem 19.9 If the fctor (x α) occurs r times then we sy it is n r-fol root, or α hs multiplicity of r. Theorem 19.10 If P (x) hs root of multiplicity m then P (x) hs root of multiplicity (m 1). Theorem 19.11 If P (x) over F is irreucible over F n eg P (x) > 1, then there re no roots of P (x) over F. Theorem 19.1 ifferent polynomils A(x) n B(x) over non-finite fiel F cnnot specify the sme function in F. Theorem 19.13 If two polynomils of the nth egree over fiel F specify the sme function for more thn n elements of F, then the two polynomils re equl.
7 Mths formul book Theorem 19.14 If P (x) = 0 + 1 x + x + + n x n over F where n 0 n if P (x) is completely reucible to n liner fctors over F (i.e., if P (x) = n (x α 1 )(x α )... (x α n ) where α 1, α,..., α n re the roots of P (x)), then α = α n 1 n (19.1) αβ = α n n (19.) αβγ = α n 3 n (19.3) Also. (19.4)... αβγ... ω = ( 1) n α 0 n (19.5) ( ) α = α αβ (19.6) Definition of rtionl function: Definition 19. If P (x), Q(x) re polynomils over F, then P (x) Q(x) rtionl function over F. Definition of sum n prouct of rtionl functions: Definition 19.3 is A(x) B(x) + C(x) A(x)D(x) + C(x)B(x) = D(x) B(x)D(x) A(x) B(x) C(x) D(x) = A(x)C(x) B(x)D(x) (19.7) (19.8) 19. Prtil frctions Theorem 19.15 If F, P, Q re reltively prime polynomils over F n eg F < eg P Q then we cn fin unique polynomils A, B such tht F P Q = A P + B Q eg A < eg P, eg B < eg Q. (19.9)