1 Algera and Trigonometry Notes on Topics that YOU should KNOW from your prerequisite courses! Here is a general Factoring Strategy that you should use to factor polynomials. 1. Always factor out the GCF(Greatest Common Factor) first.. Net check the numer of terms in your polynomial. A. Two terms (KNOW all three of these formulas!) i. Factor the difference of two squares a a a ii. Factor the difference of two cues a a a a 3 3 iii. Factor the sum of two cues a a a a 3 3 B. Three terms ---- try reverse foil (or some other method that YOU are good at) (Note: sometimes a three term polynomial will factor into the product of two trinomials) C. Four terms ---- try factor y grouping D. For five or more terms OR if none of the aove work you could try to use the rational roots theorem(from Precalculus/College Algera) or Newton s Method (which we will learn later in this course)! 3. Repeat step until all factors are prime. ALL CALCULUS STUDENTS SHOULD BE ABLE TO FACTOR EXPRESSIONS (not always just polynomials) IN A QUICK AND ACCURATE MANNER.
Properties of fractions: PROPERTIES you should already know. 1. a c ac d d. 3. a a c ac c c a c a d d c c 0 Do YOU know why this is? 4. a c ad c d d 5. a a c c c 6. a a a c c c We will e using a variety of different properties of fractions in Calculus. You should know how to comine fractions, split fractions up and simplify fractions using legitimate mathematical properties. DO NOT INVENT YOUR OWN PROPERTIES THAT DON T WORK!!! Graphs of Basic relations. You need to KNOW what the graphs of some asic relations look like from your College Algera + Trig classes (or Precalculus class). In particular you should know WHEN (if ever) these functions are zero. See the net couple of pages for the graphs!!! 1 ln 1 sin cos tan 3 y y y y y y y e y y y y NOTE: You should also rememer the transformations that you were taught in College Algera / Precalculus as well!!!!
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5 You should have learned aout the rational roots theorem (and how to use synthetic division to assist you in finding zeros of polynomials) Rational Roots Theorem: f a a a... a a a e a polynomial with integer coefficients. n n1 n Let n n1 n 1 0 If the polynomial has any rational zeros (roots), p/q, then p must e an integer factor of a 0 and q must e a factor of a n. Eample: List the possile rational zeros for 4 3 p : 1,, 4 q : 1, 3 p 1 4 : 1,,,, 4, q 3 3 3 f 3 11 10 4. Other important polynomial theorems for College Algera / Precalculus. Conjugate Pairs Theorems. i. If your polynomial has rational coefficients and a c is a zero then so is it s conjugate a c ii. If your polynomial has real coefficients and a i is a comple zero then so is it s conjugate a i A) The Remainder Theorem. If you wish to evaluate a polynomial at a numer c just do synthetic division using c and whatever remainder you get will e f (c). Note: This works for ANY numer, integer, irrational or imaginary. B) The Factor Theorem. If doing synthetic division with c yields a remainder of zero then we say that c is a zero (or root) of f () AND it means that ( c ) is a factor of f (). C) The Upper Bound Theorem If doing synthetic division with a positive numer yields a whole row of non-negatives then there is no zero greater than the one that you just tried. D) The Lower Bound Theorem If doing synthetic division with a negative numer yields a whole row of alternating signs then there is no zero smaller than the one that you just tried. E) The Intermediate Value Theorem. For any polynomial P(), with real coefficients, if a is not equal to and if P(a) and P() have opposite sings (one negative and one positive) then P() MUST have at least one zero in the interval (a, ). Note: The Intermediate Value Theorem holds for any CONTINUOUS function. We will study the idea of continuity in MAT0.
You should also recall the properties of logarithm as we will need to use some of these later on in the calculus course. I have NOT provided any logarithm prolems to practice at this point ut if you want to practice some just go to section.8 of my MAT187 Precalculus course. Definition of Logarithm: y Log (Note: is a positive real numer, 1. Logarithms can only e fed positive numers) Properties of Logarithm: (keep in mind the domain of the log function) 1. Log 1 y 6. Log 1 0 p 3. Log p 4. Product Property Log ( mn) Log m Log n m 5. Quotient Property Log Log m Log n n p 6. Power Property Log m plog m 7. One-to-One Property Log m Log n implies that m = n 8. Logarithm of each side. m = n implies Log m Log n (also note: y y ) Log p 9. Inverse Property p (also note: Log ) 10. Change of Base Formula Log Log Log a a WE WILL NEED YOU TO KNOW THIS MATERIAL FOR LATER ON IN THE MAT0 COURSE!
7 Here is a copy of The Unit Circle YOU should know this completely!!!!! As YOU know from your Trigonometry ackground, the coordinate of a point on the unit circle is the cosine of the given angle and the y coordinate of a point on the unit circle is the sine of the given angle. In terms of the coordinates on the unit circle we know. y cos sin y tan 1 1 sec csc cot y y Note: the equation of the unit circle is y 1 and as is the case with every graph, if a point lies on the graph then the coordinates of the point must make the equation true. So, if you take any of the coordinates shown on the graph and sustitute them into the equation y 1 you will get a true statement. You should e ale to evaluate trigonometric epressions for common angles and inverse trigonometric functions for common numers y utilizing the unit circle in your head!
Here are 8 trigonometric identities that you studied in your Trigonometry or Precalculus class. You do NOT need to memorize all 8 of them BUT there are several that you should know ecause they come up frequently in Calculus. Trigonometric Identities 8 1. 1 Sin. Csc 1 Cos 3. Sec 1 Tan Cot 4. Sin Sin 5. Cos Cos 6. Tan Tan 7. Sin Tan 8. Cos Cot Cos Sin 9. Cos Sin 1 10. Cot 1 Csc 11. 1 Tan Sec 1. CosA B CosACosB SinASinB 13. CosA B CosACosB SinASinB 14. SinA B SinACosB CosASinB 15. SinA B SinACosB CosASinB 16. TanA B TanA TanB 1 TanATanB 18. Sin Cos 17. TanA B TanA TanB 1 TanATanB 19. Cos Sin note: # s 18 and 19 also hold for the function pairs.tan, cot AND sec, csc 0. Sin SinCos 1. Cos Cos Sin Cos 1 =1 Sin Tan. Tan 1 Tan 3. 1 Cos Sin 4. 1 Cos Cos 5. 1 Cos Tan 1 Cos 6. Sin 1 Cos 7. Cos 1 Cos 8. Tan 1 Cos 1 Cos = Sin 1 Cos = 1 Cos Sin The identities that come up often in calculus are # s 1 3, 7, 9 11, 1 15, 0, 1. Note: # s 4 6 state that our trigonometric functions are either odd or even functions! A function is even if f f for all in the domain of the function. A function is odd if f f for all in the domain of the function. Those of you going on to take Calc II will also need to know the power reducing identities # s 3 and 34
9 When you studied Trigonometry you restricted the domain on your trigonometric functions so they ecame 1-to-1 and consequently would have an inverse that was also a function. 1 1 f sin f sin 1 1 1 y1 y f f 1 1 cos 0 cos 1 1 1 y 1 0 y y y 1 1 f tan f tan y Knowing the RANGE for the inverse trigonometric functions will e very important later in this class. REMEMBER THAT IF YOU NEED TO REVIEW MORE TRIGONOMETRY THEN YOU CAN ALWAYS GO TO MY MAT187 PRECACLULUS PAGE AND CHECK IN THE HELP SECTION TOWARRDS THE BOTTOM FOR VARIOUS TRIGONOMETRY TOPICS. Keep this packet handy as you may need to refer to it throughout this course!!!!