Math 130 Midterm Review

Transcription

1 Mth 130 Midterm Review April 6, Topic Outline: The following outline contins ll of the mjor topics tht you will need to know for the exm. Any topic tht we ve discussed in clss so fr my pper on the exm, but most questions will be bsed on items from this list. Note: the exm will focus on chpters 4-6, but you my still need to pply skills from chpters 1-3 s well. 1. Pre-Algebr () Numericl Expressions nd Algebric Expressions (b) Algebric Techer s Solutions (c) Solving Bsic Equtions (d) Algebric Properties of Arithmetic i. Commuttive Lws: [ + b = b + nd b = b] ii. Associtive Lws: [( + b) + c = + (b + c) nd (b)c = (bc)] iii. Distributive Lw: [(b + c) = b + c] iv. Additive Identity: [ + 0 = ] v. Multiplictive Identity: [ 1 = ] (e) Deriving Algebric Identities (such s ( + b)( b) = 2 b 2, ( + b) 3 = b + 3b 2 + b 3, &c.) (f) Exponents i. Definition ( m = } {{ } ) mtimes ii. Properties 1

2 A. m n = m+n B. m = m n (if m n nd 0) n C. ( m ) n = mn D. m b m = (b) m E. 0 = 1 if 0 iii. Simplifying expressions involving exponents (especilly frctions) iv. Cnnot define Fctors, Primes, nd Proofs () Proofs involving the sum nd product of even or odd numbers (b) Divisibility i. Definition: A is divisible by k (or k divides A ) mens there is some whole number such tht A = k, i.e. A is multiple of k). ii. Divisibility Lemm iii. Divisibility tests for 2, 3, 4, 5, 6, 8, 9, 10, nd 11 A. Applying these tests to check divisibility B. Proving these divisibility tests (t lest for 4-digit numbers) (c) Primes i. Definitions of prime nd composite numbers ii. Fundmentl Theorem of Arithmetic: Every whole number cn be fctored into product of primes, nd this fctoriztion is unique up to re-ordering. iii. Writing prime fctoriztions (especilly in exponentil form) iv. Testing whether whole number is prime A. Using the Primlity Test : A whole number, N, is prime unless it s smllest prime fctor, p, stisfies p 2 N. (d) GCD nd LCM i. Definitions of GCD nd LCM ii. Finding GCD nd LCM for pir of whole numbers (possibly written s lgebric expressions) A. By listing fctors nd multiples B. Using prime fctoriztion C. Using Euclid s Algorithm 2

3 3. Frctions iii. GCD(, b) LCM(, b) = b () Definitions (b) Common digrms used to represent frctions i. Are Models: (pie digrms, br digrms, &c.) ii. Number line iii. Set Model (c) Rules of Frction Arithmetic: i. Equivlent Frctions: = n for ny whole number n 0 b bn ii. Addition: + c = d+bc (If we hve common denomintor, b d bc + c = +c) b b b (If we hve common denomin- iii. Subtrction: c = d bc b d bc tor, c = c) b b b iv. Multipliction: c = c b d bd v. Frctions nd whole number division: b = b vi. Division: = d b b c d denomintors nd using b c b = c ) (d) Compring the size of two frctions (e) Computing with frctions (f) Frction division word problems c (This results from finding common i. Br digrms representing both prtitive nd mesurement division ii. Writing one nd two step word problems (for ech model of division) iii. Writing techer s solutions to one nd two step word problems (for ech model of division) (g) Frctions s step towrds lgebr: i. Rules of frction rithmetic (given bove) ii. We gin the multiplictive inverse property: For ech frction x there is some multiplictive inverse for x, written 1 x, such tht x 1 x = 1. (If x is b, then 1 x is b.) iii. Writing step-by-step solutions to frction problems, with pproprite justifictions for ech step 3

4 2 Smple Problems The following problems re intended to help you test your knowledge of the mjor topics on the exm. They do not mtch the ctul questions on the exm (lthough I hope tht nyone who is comfortble solving these problems will lso do well on the exm). These questions re orgnized roughly in the order tht the topics pper in the bove list. Problem 1. Derive the following lgebric identities (using only the lgebric properties of rithmetic given bove, not FOIL): 1. ( 2b) 2 = 2 4b + 4b 2 2. ( + b c) 2 = 2 + b 2 + c 2 + 2b 2c 2bc 3. ( + b) 4 = b b 2 + 4b 3 + b 4 4. ( b)( 2 + b + b 2 ) = 3 b 3 Problem 2. Simplify the following expressions s much s possible, leving your nswer in exponentil form: 1. 6x 4 y 2 8xy (2b) 2 (b 2 ) 3 (3b) 2 ( 4 b) 2 Problem 3. Prove, using the definition of exponentition nd the rules of rithmetic, tht n b n = (b) n. Problem 4. Give the prime fctoriztions for ech of the following numbers, nd if ny re prime, explin why (simply sying tht number N hs no fctors other thn 1 nd N is not sufficient, you must show tht this is true): ! 6. 5! 7! Problem 5. Prove the divisibility test for division by 8. You my use the Divisibility Lemm. 4

5 Problem 6. Find the GCD nd LCM of ech of the following pirs of numbers (you should be ble to do this with both prime fctoriztion nd Euclid s Algorithm): 1. (1960, 2310) 2. (1323, 7203) Problem Write one step word problem, using mesurement division, tht uses 2 3 in its solution Write one step word problem, using prtitive division, tht uses in its solution. 3. Write two step word problem, using prtitive division, tht uses in its solution. 4. Write techer s solutions to ech of the word problems tht you wrote. Problem 8. Write techer s solution for the following word problem (you should be ble to write both n lgebric solution nd non-lgebric solution), mking sure to drw pproprite br digrms: Bob spent 1 of his money on ht nd then spent 2 of his remining 5 3 money on shoes. If he hd 24 dollrs left, how much did he spend on the ht? Problem 9. Simplify the following expression, justifying ech step (try doing this with the distributive lw nd without the distributive lw): (( ) 5 ) Problem 10. Prove tht the sum of four consecutive whole numbers is lwys even. Problem 11. Which frction is lrger, your inequlity holds for ll n > 0. n 2n+1 n+1 or? Be sure to justify tht 2n 1 5