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ALGEBRA Factors and Zeros of Polynomials Let px an x n an1x n1 . . . a1x a0 be a polynomial. If pa 0, then a is a zero of the polynomial and a solution of the equation px 0. Furthermore, x a is a factor of the polynomial. Fundamental Theorem of Algebra An nth degree polynomial has n (not necessarily distinct) zeros. Although all of these zeros may be imaginary, a real polynomial of odd degree must have at least one real zero. Quadratic Formula If px ax 2 bx c, and 0 ≤ b2 4ac, then the real zeros of p are x b ± b2 4ac2a. Special Factors x 2 a 2 x ax a x 3 a 3 x ax 2 ax a 2 x 3 a3 x ax 2 ax a 2 x 4 a 4 x 2 a 2x 2 a 2 Binomial Theorem x y2 x 2 2xy y 2 x y2 x 2 2xy y 2 x y3 x 3 3x 2y 3xy 2 y 3 x y3 x 3 3x 2y 3xy 2 y 3 x y4 x 4 4x 3y 6x 2y 2 4xy3 y 4 x y4 x 4 4x 3y 6x 2y 2 4xy 3 y 4 nn 1 n2 2 . . . x y nxy n1 y n 2! nn 1 n2 2 . . . x yn x n nx n1y x y ± nxy n1 y n 2! x yn x n nx n1y Rational Zero Theorem If px an x n a n1x n1 . . . a1x a0 has integer coefficients, then every rational zero of p is of the form x rs, where r is a factor of a0 and s is a factor of an. Factoring by Grouping acx 3 adx 2 bcx bd ax 2cx d bcx d ax 2 bcx d Arithmetic Operations ab ac ab c ab a d ad c d b c bc b ab a c c a c ad bc b d bd a b a c bc ab a b c c c ab ba cd dc ab ac bc a a ac b b c Exponents and Radicals a0 1, ab x a0 ax bx ab x a xb x a xa y a xy n am amn ax © Houghton Mifflin Company, Inc. 1 ax a a12 ax a xy ay n a a1n n n a n b ab axy a xy ab n n a n b DERIVATIVES AND INTEGRALS Basic Differentiation Rules 1. 4. 7. 10. 13. 16. 19. 22. 25. 28. 31. 34. d cu cu dx d u vu uv dx v v2 d x 1 dx d u e eu u dx d sin u cos uu dx d cot u csc2 uu dx d u arcsin u dx 1 u2 d u arccot u dx 1 u2 d sinh u cosh uu dx d coth u csch2 uu dx d u sinh1 u dx u2 1 d u coth1 u dx 1 u2 2. 5. 8. 11. 14. 17. 20. 23. 26. 29. 32. 35. d u ± v u ± v dx d c 0 dx d u u u , u 0 dx u d u loga u dx ln au d cos u sin uu dx d sec u sec u tan uu dx d u arccos u dx 1 u2 d u arcsec u dx u u2 1 d cosh u sinh uu dx d sech u sech u tanh uu dx d u cosh1 u dx u2 1 d u sech1 u dx u1 u2 3. 5. 7. 9. 11. 13. 15. 17. kf u du k f u du 2. du u C 4. eu du eu C 6. cos u du sin u C 8. cot u du ln sin u C 10. csc u du ln csc u cot u C 12. csc2 u du cot u C 14. csc u cot u du csc u C 16. du 1 u arctan C 2 u a a 18. a2 © Houghton Mifflin Company, Inc. 6. 9. Basic Integration Formulas 1. 3. 12. 15. 18. 21. 24. 27. 30. 33. 36. d uv uv vu dx d n u nu n1u dx d u ln u dx u d u a ln aau u dx d tan u sec2 uu dx d csc u csc u cot uu dx d u arctan u dx 1 u2 d u arccsc u dx u u2 1 d tanh u sech2 uu dx d csch u csch u coth uu dx d u tanh1 u dx 1 u2 u d csch1 u dx u 1 u2 f u ± gu du au du ln1aa u f u du ± C sin u du cos u C tan u du ln cos u C sec u du ln sec u tan u C sec2 u du tan u C sec u tan u du sec u C du u C a du 1 u arcsec C 2 2 a a uu a a2 u2 arcsin gu du FORMULAS FROM GEOMETRY Triangle Sector of Circular Ring h a sin 1 Area bh 2 (Law of Cosines) c p average radius, w width of ring, in radians Area pw a θ h b c2 a 2 b2 2ab cos c (Pythagorean Theorem) a2 Area ab a b2 Equilateral Triangle Area s h 4 A Right Circular Cone r 2h 3 Lateral Surface Area rr2 h2 h h Volume b h Area a b 2 h s Parallelogram Trapezoid a A area of base Ah Volume 3 s 3s2 Area bh a 2 b2 2 Cone 3s 2 r Frustum of Right Circular Cone a h a h r rR h 3 Lateral Surface Area sR r Volume b b r2 R2 s h Right Circular Cylinder Circle r2 Area Circumference 2 r Volume r 2h Lateral Surface Area 2 rh r Sector of Circle in radians r2 Area 2 s r w b Circumference 2 b h θ Ellipse Right Triangle c2 p R r h Sphere 4 Volume r 3 3 Surface Area 4 r 2 s θ r Circular Ring p average radius, w width of ring Area R 2 r 2 2 pw © Houghton Mifflin Company, Inc. r Wedge r p R w A area of upper face, B area of base A B sec A θ B TRIGONOMETRY Definition of the Six Trigonometric Functions Opposite Right triangle definitions, where 0 < < 2. opp hyp sin csc e s u hyp opp en pot Hy adj hyp cos sec θ hyp adj Adjacent opp adj tan cot adj opp Circular function definitions, where is any angle. y r y sin csc r = x2 + y2 r y (x, y) x r r cos sec θ y r x x y x x cot tan x y Reciprocal Identities 1 sin x csc x 1 csc x sin x 1 sec x cos x 1 cos x sec x sin x cos x cot x cos x sin x Pythagorean Identities sin2 x cos2 x 1 1 tan2 x sec2 x (− 12 , 23 ) π (0, 1) ( 12 , 23 ) 90° (− 22 , 22 ) 3π 23π 2 π3 π ( 22 , 22 ) 120° 60° 4 π 45° (− 23 , 12) 56π 4150°135° ( 23 , 21) 6 30° (− 1, 0) π 180° 210° 330° 1 cot2 x csc2 x sin 2u 2 sin u cos u cos 2u cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u 2 tan u tan 2u 1 tan2 u Power-Reducing Formulas 1 cos 2u 2 1 cos 2u 2 cos u 2 1 cos 2u tan2 u 1 cos 2u sin2 u Sum-to-Product Formulas 2 x cos x csc x sec x 2 sec x csc x 2 sin u sin v 2 sin 2 x sin x tan x cot x 2 cot x tan x 2 cos Reduction Formulas sinx sin x cscx csc x secx sec x x (− 23 , − 12) 76π 5π 225°240° 300°315°7π 116π ( 23 , − 21) (− 22 , − 22 ) 4 43π 270° 32π 53π 4 ( 22 , − 22 ) 1 3 (0, − 1) ( 2 , − 2 ) (− 12 , − 23 ) Cofunction Identities sin 0° 0 360° 2π (1, 0) Double -Angle Formulas 1 tan x cot x 1 cot x tan x Tangent and Cotangent Identities tan x y cosx cos x tanx tan x cotx cot x Sum and Difference Formulas sinu ± v sin u cos v ± cos u sin v cosu ± v cos u cos v sin u sin v tan u ± tan v tanu ± v 1 tan u tan v © Houghton Mifflin Company, Inc. u 2 v cosu 2 v uv uv sin u sin v 2 cos sin 2 2 uv uv cos u cos v 2 cos cos 2 2 uv uv cos u cos v 2 sin sin 2 2 Product-to-Sum Formulas 1 sin u sin v cosu v cosu v 2 1 cos u cos v cosu v cosu v 2 1 sin u cos v sinu v sinu v 2 1 cos u sin v sinu v sinu v 2