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- 1. Elementary Algebra SHEILA MARIEC. NAVARRO
- 2. REAL NUMBERS SYSTEM RealNumber (R) Rationals (Q) Integers (Z) Negative Integers (Z-) Whole Number (W) {0} Positive Integers (Z+) Prime No. (P) {1} Composite No. (C) Even No. (E) Odd No. (O) Fractions (F) Irrationals (Q’) Notes: 1. Real numbers are composed of rational and irrational numbers. 2. Rational numbers are either terminating or repeating decimals, otherwise it is called irrational. 3. Zero (0) is neither positive nor negative. 4. One (1) is not a prime no. 5. If an integer is not odd, then it is even. 6. 6.4 is not an even number.
- 3. Question: 1. What isthe sum of the smallest prime and largest 2-digit prime number? a. 95 b. 98 c. 99 d. 100 ANS: C
- 4. Question: Find the sumof all odd integers from -100 to 110? a. 520 b. 525 c. 530 d. 540 ANS: B -100+(-99)+…+99+100=0 101+103+105+107+109=525
- 5. SETS Union (“or” /∪) All elements of two sets 𝑨 ∪ 𝑩 = {𝟏, 𝟑, 𝟒, 𝟓, 𝟕, 𝟖} Intersection (“and” / ∩) All common elements of two sets 𝐴 ∩ 𝐵 = {1} Complement ( ‘/ (𝑈 − 𝐴)) Elements in 𝑈 not in 𝐴 𝐴′ = {2, 3, 6, 7, 8, 9, 10} Difference (− / (𝐴 − 𝐵)) Elements in 𝐴 not in 𝐵 𝐴 − 𝐵 = {4, 5} Cross Product (𝐴 × 𝐵) Ordered pairs with domain from 𝐴 and abscisa from 𝐵 𝐴 × 𝐵 = { 1,1 , 1, 3 , … , 5, 7 , 5,8 ) A = {1, 4, 5} B = {1, 3, 7, 8} U = {1, 2, …, 10}
- 6. Question: Which of thefollowing statement is true? a. If 𝑄 is the universal set, then 𝐹′ = 𝑍. b. 𝑄′ − 𝑄 = { } c. 𝑄 ∩ 𝑍 = 𝐹 d. 𝑍 − 𝑍− = 𝑍+ ANS: A
- 7. Question: During a survey,the following data were gathered: 60 students like algebra, 50 students like calculus, and 45 likes physics. Thirty students like both algebra and calculus; 25 students like both calculus and physics; and 20 students like algebra and physics. Only 15 students like all the three subjects. How many students were surveyed? a. 155 b. 125 c. 115 d. 95 ANS: D Algebra Calculus Physics 15 5 10 15 25 10 15 25+5+15+15+10+10+15=95
- 8. SOME THEOREMS ONREAL NUMBERS • Theorem 1. − 𝑎 𝑏 = −𝑎 𝑏 = 𝑎 −𝑏 . • Theorem 2. 𝑎𝑐 𝑏𝑐 = 𝑎 𝑏 . • Theorem 3. 𝑎 𝑐 + 𝑏 𝑐 = 𝑎+𝑏 𝑐 or 𝑎 𝑐 + 𝑏 𝑑 = 𝑎𝑑+𝑏𝑐 𝑐𝑑 . • Theorem 4. 𝑎 𝑏 × 𝑐 𝑑 = 𝑎𝑐 𝑏𝑑 . • Theorem 5. 𝑎 𝑏 ÷ 𝑐 𝑑 = 𝑎 𝑏 × 𝑑 𝑐 = 𝑎𝑑 𝑏𝑐 . NOTE: In the following theorems, 𝑎, 𝑏, 𝑐, 𝑑 may stand for any algebraic expression. 0 𝑎 = 0; 𝑎 0 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑; 𝑎 ∞ = 0. Common Mistake • 𝑥2−3𝑥+2 𝑥2−2𝑥+1 = (𝑥−2)(𝑥−1) (𝑥−1)(𝑥−1) = 𝑥−2 𝑥−1 • 𝑥−2 𝑥+1 𝑥−2 𝑥−3 = 𝑥−2 𝑥+1 𝑥−2 𝑥−3
- 9. Fundamental Operations • Additionand Subtraction is only applicable to similar terms. 1. 2𝑥 − 3𝑦 + (5𝑥 + 2𝑦) = 2. 2𝑥 − 3𝑦 − 5𝑥 + 2𝑦 = • For Multiplication and Division, laws of exponents are needed. 𝟕𝒙 − 𝒚 2𝑥 − 3𝑦 − 5𝑥 − 2𝑦 = −𝟑𝒙 − 𝟓𝒚
- 10. Laws of Exponents 1.𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 2. 𝑎𝑚 𝑎𝑛 = 𝑎𝑚−𝑛 3. 𝑎𝑚 𝑛 = 𝑎𝑚𝑛 4. 𝑎𝑏 𝑚 = 𝑎𝑚 𝑏𝑚 5. 𝑎 𝑏 𝑚 = 𝑎𝑚 𝑏𝑚 6. 𝑎 𝑚 𝑛 = 𝑛 𝑎𝑚 7. 𝑎0 = 1 provided 𝑎 ≠ 0. Note: 00 is indeterminate. 6. 𝑎−𝑚 = 1 𝑎𝑚 7. If 𝑎𝑚 = 𝑎𝑛, then 𝑚 = 𝑛.
- 11. Laws of Radicals 1.𝑎 1 𝑛 = 𝑛 𝑎 2. 𝑎 𝑚 𝑛 = 𝑛 𝑎𝑚 = 𝑛 𝑎 𝑚 3. 𝑛 𝑎 𝑛 = 𝑎 4. 𝑛 𝑎 × 𝑛 𝑏 = 𝑛 𝑎𝑏 5. 𝑛 𝑎 𝑛 𝑏 = 𝑛 𝑎 𝑏 provided that 𝑏 ≠ 0 Illustration: 1. 2𝑥 3 = 23 𝑥3 = 𝟖𝒙𝟑 2. 𝑥3 𝑦 2 = 𝑥3 2 𝑦2 = 𝒙𝟔 𝒚𝟐 3. 2𝑥𝑦3 4 = 2𝑥𝑦3 4 2 = 22 𝑥2 𝑦3 2 = 𝟒𝒙𝟐 𝒚𝟔
- 12. Question: 𝑎 + 𝑏𝑏 ∙ 𝑎 − 𝑏 𝑏 is equivalent to? a. 𝑎 + 𝑏 𝑏 b. 𝑎2 − 𝑏3 c. 𝑎 − 𝑏 𝑏 d. 𝑎2 − 𝑏2 ANS: B 𝑎 + 𝑏 𝑏 𝑎 − 𝑏 𝑏 𝑎2 − 𝑏2 𝑏 𝒂𝟐 − 𝒃𝟑
- 13. Question: Simplify: 𝑎 + 1 𝑎 2 −𝑎 − 1 𝑎 2 . a. −4 b. 0 c. 4 d. − 2 𝑎2 ANS: C 𝑎 + 1 𝑎 + 𝑎 − 1 𝑎 𝑎 + 1 𝑎 − 𝑎 − 1 𝑎 2𝑎 2 𝑎 𝟒
- 14. Question: Simplify: 𝑥2𝑦3𝑧−2 −3 𝑥−3𝑦𝑧3 − 1 2 𝑥𝑦𝑧−3− 5 2 . a. 1 𝑥2𝑦7𝑧3 b. 1 𝑥2𝑦7𝑧5 c. 1 𝑥2𝑦5𝑧3 d. 1 𝑥5𝑦7𝑧3 ANS: A = 𝑥−6 𝑦−9 𝑧6 𝑥 3 2𝑦− 1 2𝑧− 3 2 𝑥 − 5 2 𝑦− 5 2𝑧 15 2 = 𝑥−6𝑦−9𝑧6 𝑥−4𝑦−2𝑧9 = 𝟏 𝒙𝟐𝒚𝟕𝒛𝟑
- 15. Special Products andFactoring 𝑎𝑥 + 𝑎𝑦 = 𝑎(𝑥 + 𝑦) Multinomial Common Factor 𝑥2 − 𝑦2 = 𝑥 + 𝑦 𝑥 − 𝑦 Difference of two squares 𝑥3 ± 𝑦3 = (𝑥 ± 𝑦)(𝑥2 ∓ 𝑥𝑦 + 𝑦2) Sum/Diff. of two cubes 𝑥2 ± 2𝑥𝑦 + 𝑦2 = 𝑥 ± 𝑦 2 Perfect Square Trinomials 𝑎𝑐𝑥2 + 𝑎𝑑 + 𝑏𝑐 𝑥 + 𝑏𝑑 = (𝑎𝑥 + 𝑏)(𝑐𝑥 + 𝑑) FOIL/Trial-and-Error 𝒓𝒕𝒉 𝒕𝒆𝒓𝒎 = 𝒏! 𝒏 − 𝒓 + 𝟏 ! 𝒓 − 𝟏 ! 𝒂𝒏−𝒓+𝟏 𝒃𝒓−𝟏 Where 𝑟 is the unknown terms and 𝑛 is the total number of unknown terms If 𝑏2 − 4𝑎𝑐 is a perfect square the quadratic trinomial is factorable.
- 16. Horizontal Multiplication We canalways multiply polynomials by polynomials using the distribution methods. Multiply (2𝑥2 − 3𝑥𝑦 + 𝑦2 )(−3𝑥2 + 2𝑥𝑦 − 3𝑦2 ). −3𝑥2 2𝑥2 − 3𝑥𝑦 + 𝑦2 + 2𝑥𝑦 2𝑥2 − 3𝑥𝑦 + 𝑦2 − 3y2(2𝑥2 − 3𝑥𝑦 + 𝑦2) −6𝑥4 + 9𝑥3 𝑦 − 3𝑥2 𝑦2 + 4𝑥3 𝑦 − 6𝑥2 𝑦2 + 2𝑥𝑦3 − 6𝑥2 𝑦2 + 9𝑥𝑦3 − 3𝑦4 −𝟔𝒙𝟒 + 𝟏𝟑𝒙𝟑 𝒚 − 𝟏𝟓𝒙𝟐 𝒚𝟐 + 𝟏𝟏𝒙𝒚𝟑 − 𝟑𝒚𝟒
- 17. Division of Polynomials •Divide 𝑥4 − 18𝑥2 + 32 by 𝑥 − 4 𝑥3 + 4𝑥2 − 2𝑥 − 8 𝑥 − 4 𝑥4 + 0𝑥3 − 18𝑥2 + 0𝑥 + 32 𝑥4 − 4𝑥3 4𝑥3 − 18𝑥2 4𝑥3 − 16𝑥2 −2𝑥2 + 0𝑥 −2𝑥2 + 8𝑥 −8𝑥 + 32 −8𝑥 + 32 0 1 0 -18 0 32 |4 4 16 -8 -32 1 4 -2 -8 0 Long Division Synthetic Division
- 18. Factor and Remaindertheorem Given a function 𝑓(𝑥) and a binomial 𝑥 − 𝑐 where c is constant. FACTOR. If 𝑓 𝑐 = 0, then 𝑥 − 𝑐 is a factor of 𝑓(𝑥). Example: f x = 2𝑥3 + 5𝑥2 − 𝑥 − 6 and 𝑥 + 2. 𝑓 −2 = 2 −2 3 + 5 −22 − −2 − 6 = 0 REMAINDER. If 𝑓 𝑐 = 𝑧 where 𝑧 ≠ 0, then 𝑧 is the exponent whenever 𝑓(𝑥) is divided by 𝑥 − 𝑐. Example: 𝑓 2 = 2 23 + 5 22 − 2 − 6 = 28
- 19. Sum of Coefficientof Variables Substitute 1 to each variables in the polynomials then simplify the expression to get the sum of its coefficients. Example: Find the coefficients of the variables in the expansion 2𝑥 + 3𝑦 − 1 4. 2 1 + 3 1 − 1 4 = 44 = 256
- 20. Question: If 𝑥2 + 𝑦2 =22 and 𝑥𝑦 = 9, find the value of 𝑥 − 𝑦 2 ? a. 4 b. 13 c. 31 d. 40 ANS: A 𝑥 − 𝑦 2 = 𝑥2 − 2𝑥𝑦 + 𝑦2 = 22 − 2 9 = 4
- 21. Question: Factor 𝑎4 − 𝑏2 +𝑏 − 𝑎2 as completely as possible. a. 𝑎2 + 𝑏 𝑎2 + 𝑏 − 1 b. 𝑎2 + 𝑏 𝑎2 − 𝑏 − 1 c. 𝑎2 − 𝑏 𝑎2 + 𝑏 − 1 d. 𝑎2 − 𝑏 𝑎2 − 𝑏 − 1 ANS: C = 𝑎4 − 𝑏2 − 𝑎2 − 𝑏 = 𝑎2 + 𝑏 𝑎2 − 𝑏 − 𝑎2 − 𝑏 = (𝒂𝟐 − 𝒃)(𝒂𝟐 + 𝒃 − 𝟏)
- 22. Question: Which of thefollowing are the correct factors of 6𝑥2 + 23𝑥 − 4? a. 6𝑥 − 1 𝑥 + 4 b. 6𝑥 + 1 𝑥 − 4 c. 3𝑥 + 4 2𝑥 − 1 d. (3𝑥 − 4)(2𝑥 + 1) ANS: A 𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 6: 3, 2, 1, 6 𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 − 4: 1, −4, 2, −2 6𝑥 𝑥 (𝟔𝒙 − 𝟏)(𝒙 + 𝟒)
- 23. Question: Factor the expression𝑥6 − 1 as completely as possible. a. 𝑥 + 1 𝑥 − 1 𝑥4 + 𝑥2 − 1 b. 𝑥 + 1 𝑥 − 1 𝑥4 + 2𝑥2 + 1 c. 𝑥 + 1 𝑥 − 1 𝑥4 − 𝑥2 + 1 d. (𝑥 + 1)(𝑥 − 1)(𝑥4 + 𝑥2 + 1) ANS: D 𝑥2 − 1 𝑥4 + 𝑥2 + 1 (𝑥 + 1)(𝑥 − 1)(𝑥4 + 𝑥2 + 1)
- 24. Question: Give a 3rdterm so that 4𝑥4 + 9𝑦2 becomes a perfect square trinomial. a. 6𝑥2 𝑦 b. 12𝑥2𝑦2 c. 12𝑥2 𝑦 d. 36𝑥2𝑦 ANS: C 4𝑥4 + 9𝑦2 1st = 2𝑥2 2nd = 3𝑦 3rd = 2*1st *2nd = 2 2𝑥2 3𝑦 = 𝟏𝟐𝒙𝟐 𝒚
- 25. Question: Simplify: 5𝑥 2𝑥2+7𝑥+3 − 𝑥+3 2𝑥2−3𝑥−2 + 2𝑥+1 𝑥2+𝑥−6 . a. 2 𝑥−3 b. 𝑥−3 5 c. 𝑥+3 𝑥−1 d. 4 𝑥+3 ANS: D = 5𝑥 2𝑥 +1 𝑥 + 3 − 𝑥 + 3 2𝑥 + 1 𝑥 − 2 + 2𝑥 + 1 𝑥 + 3 𝑥 − 2 = 5𝑥 𝑥 − 2 − 𝑥 + 3 𝑥 + 3 + 2𝑥 + 1 2𝑥 + 1 2𝑥 + 1 𝑥 + 3 𝑥 − 2 = 5𝑥2 − 10𝑥 − 𝑥2 − 6𝑥 − 9 + 4𝑥2 + 4𝑥 + 1 2𝑥 + 1 𝑥 + 3 𝑥 − 2 = 8𝑥2 − 12𝑥 − 8 2𝑥 + 1 𝑥 + 3 𝑥 − 2 = 4 2𝑥 + 1 𝑥 − 2 2𝑥 + 1 𝑥 + 3 𝑥 − 2 = 𝟒 𝒙 + 𝟑
- 26. Question: Simplify: 𝑚3−8 2𝑚−1 ∙ 2𝑚2+3𝑚−2 𝑚2−4 . a. 𝑚2 +2𝑚 + 4 b. 𝑚2 + 2𝑚 − 4 c. 𝑚2 − 2𝑚 + 4 d. 𝑚2 − 2𝑚 − 4 ANS: A = 𝑚 − 2 𝑚2 + 2𝑚 + 4 2𝑚 − 1 ∙ 2𝑚 − 1 𝑚 + 2 𝑚 + 2 𝑚 − 2 = 𝒎𝟐 + 𝟐𝒎 + 𝟒
- 27. Question: What is theremainder if 2𝑥3 − 3𝑥2 + 5𝑥 − 4 is divided by 𝑥 − 1? a. 0 b. -8 c. -4 d. 6 ANS: A 𝑥 − 𝑐 = 𝑥 − 1 𝑠𝑜 𝑐 = 1 = 2 1 3 − 3 1 2 + 5 1 − 4 = 𝟎
- 28. Question: Given: 𝑓 𝑥= 𝑥 − 3 𝑥 + 4 + 4 when divided by (𝑥 − 𝑘), the remainder is 𝑘. Find 𝑘. a. 2 b. 3 c. 4 d. -3 ANS: C 𝑓 𝑥 = 𝑥2 + 𝑥 − 8 𝑓 𝑘 = 𝑘 𝑘 = 𝑘2 + 𝑘 − 8 0 = 𝑘2 − 8 𝒌 = ±𝟒
- 29. Question: Find the sumof the coefficients of all terms in 5𝑥2 − 3𝑦2 8 ? a. 186 b. 256 c. 512 d. 542 ANS: B Substitute 1 to all variables = 5 1 2 − 3 1 2 8 = 28 or 256
- 30. Linear Equations inOne Unknown • The ultimate goal is to transform the equation into 𝑥 = 𝑎 𝑏 , which specifies the value of x. This is called the solution or roots. No Solution Unique Solution Infinitely Many Solution Solve 𝑥+5 𝑥−5 = 1; 𝑥 ≠ 5. 𝑥 − 5 𝑥 + 5 𝑥 − 5 = 1 𝑥 − 5 𝑥 + 5 = 𝑥 − 5 5 = −5 False Statement Solve 𝑥 𝑥−1 + 4 15 = 4 5𝑥−5 + 3 5 (3) 5 𝑥 + 𝑥– 1 4 = 3 4 + 3(𝑥– 1)(3) 15𝑥 + 4𝑥– 4 = 12 + 9𝑥– 9 19𝑥 – 4 = 9𝑥 + 3 19𝑥 – 9𝑥 = 3 + 4 𝑥 = 7 10 Solve 5 2𝑥 − 1 = 2 8𝑥 − 7 − 3 2𝑥 − 3 10𝑥 − 5 = 16𝑥 − 14 − 6𝑥 + 9 10𝑥 − 5 = 10𝑥 − 5 0 = 0 True Statement
- 31. Solving Linear Equationw/ Two Unknown • Graphical Method Unique Solution Infinitely Many No Solution
- 32. Solving Linear Equationw/ Two Unknown SUBSTITUTION ELIMINATION DETERMINANTS Solve: 2𝑥 − 𝑦 = 3 2𝑥 + 3𝑦 = 7 EQN 1: 𝑦 = 2𝑥 − 3 EQN 2: 2𝑥 + 3 2𝑥 − 3 = 7 2𝑥 + 6𝑥 − 9 = 7 8𝑥 = 16 𝑥 = 2 EQN 1: 𝑦 = 2 2 − 3 = 1 Solve: 2𝑥 − 𝑦 = 3 2𝑥 + 3𝑦 = 7 2𝑥 − 𝑦 = 3 −(2𝑥 + 3𝑦 = 7) −4𝑦 = −4 𝑦 = 1 EQN 1: 2𝑥 − 1 = 3 𝑥 = 2 Solve: 2𝑥 − 𝑦 = 3 2𝑥 + 3𝑦 = 7 𝑥 = 3 −1 7 3 2 −1 2 3 = 3 3 − 7 −1 2 3 − 2 −1 = 16 8 = 2 𝑦 = 2 3 2 7 2 −1 2 3 = 2 7 − 2 3 2 3 − 2 −1 = 8 8 = 1
- 33. Question: Find the valueof 𝑥 and 𝑦 that satisfies the system of equations: 3𝑥 − 𝑦 = 6 and 9𝑥 − 𝑦 = 12. a. 𝑥 = 3; 𝑦 = 1 b. 𝑥 = 1; y = −3 c. 𝑥 = 2; 𝑦 = 2 d. 𝑥 = 4; 𝑦 = 2 ANS: B 3𝑥 − 𝑦 = 6 − 9𝑥 − 𝑦 = 12 −6𝑥 = −6 Implies 𝒙 = 𝟏. 3 1 − 𝑦 = 6 𝒚 = −𝟑
- 34. Question: If 𝑓 𝑥= 2𝑥2 − 𝑥 + 1, what is the value of 𝑓 𝑥 − 𝑓(𝑥 + 1)? a. 4x-1 b. -4x-1 c. 4x+1 d. 1-4x ANS: B = 2𝑥2 − 𝑥 + 1 − 2 𝑥 + 1 2 − 𝑥 + 1 + 1 = 2𝑥2 − 𝑥 + 1 − 2𝑥2 − 4𝑥 − 2 + 𝑥 + 1 − 1 = −𝟒𝒙 − 𝟏
- 35. Solving Quadratic Equation FACTORINGQUADRATIC FORMULA ROOTS - The goal is to factor the equation completely. Make sure the factors are on left-hand side and zero (0) on right-hand side. For quadratic eqn 𝑎𝑥2 + 𝑏𝑥 + 𝑐; 𝒙 = −𝒃± 𝒃𝟐−𝟒𝒂𝒄 𝟐𝒂 Sum : 𝒓𝟏 + 𝒓𝟐 = − 𝒃 𝒂 Product: 𝒓𝟏 ∙ 𝒓𝟐 = 𝒄 𝒂 Solve: 3𝑥2 − 5𝑥 − 2 = 0 3𝑥 + 1 𝑥 − 2 = 0 3𝑥 + 1 = 0 | 𝑥 − 2 = 0 𝑥 = − 1 3 or 𝑥 = 2 Solve: 3𝑥2 − 5𝑥 − 2 = 0 𝑥 = − −5 ± −5 2 − 4 3 −2 2 3 𝑥 = 5 ± 49 6 𝑥 = − 1 3 or 𝑥 = 2 One root of the eqn 3𝑥2 − 5𝑥 + 𝑐 = 0 is 2. Find the other root and the value of 𝑐. Sum: 𝑟 + 2 = 5 3 → 𝑟 = −1/3 Product: 2 − 1 3 = 𝑐 3 → 𝑐 = −2
- 36. Remarks: 1. 𝑏2 − 4𝑎𝑐is called the discriminant of the quadratic equation. 2. If 𝑏2 − 4𝑎𝑐 = 0, then the roots are real and both are equal to − 𝑏 2𝑎 3. If 𝑏2 − 4𝑎𝑐 > 0, then the roots are real and unequal. 4. If 𝑏2 − 4𝑎𝑐 < 0, then the roots are imaginary numbers.
- 37. Question: The values of𝑥 in the equation 𝑥 2 5 − 𝑥 1 5 − 2 = 0 are: a. 1 & -2 b. -1 & 2 c. -1 & 32 d. 1 & -32 ANS: C 𝑥 1 5 2 − 𝑥 1 5 − 2 = 0 By factoring, 𝑥 1 5 − 2 𝑥 1 5 + 1 = 0 𝑥 1 5 − 2 = 0 or 𝑥 1 5 + 1 = 0 𝒙 = 𝟐𝟓 = 𝟑𝟐 or 𝒙 = −𝟏 𝟓 = −𝟏
- 38. Question: Give the sumof the roots of 2𝑥2 − 8𝑥 + 1 = 0. a. 4 b. -5 c. -2 d. 2 ANS: A 𝑟1 + 𝑟2 = − −8 2 = 𝟒
- 39. Question: Find 𝑘 inthe equation 4𝑥2 + 𝑘𝑥 + 9 = 0 so that it will only have one real root. a. 10 b. 11 c. 12 d. 13 ANS: C Real and unique roots if 𝑏2 − 4𝑎𝑐 = 0 𝑘2 − 4 4 9 = 0 𝑘2 = 144 𝒌 = ±𝟏𝟐
- 40. Question: Find the rootsof 𝑥2 − 4 + 4𝑥 = 0. a. 2 b. 2 and -2 c. 0 d. no root ANS: D 𝑥2 − 4 = − 4𝑥 𝑥2 − 4 = 4𝑥 𝑥2 − 4𝑥 − 4 = 0 𝑥 − 2 2 = 0 𝑥 = 2 Checking: 0 + 16 ≠ 0. False statement → No roots
- 41. Partial Fractions Illustrations: • 3 𝑥−1 𝑥+2 = 𝐴 𝑥−1 + 𝐵 𝑥+2 • 3𝑥2+2𝑥+1 𝑥−13 = 𝐴 𝑥−1 + 𝐵 𝑥−1 2 + 𝐶 𝑥−1 3 • 3𝑥−5 𝑥−1 𝑥2+𝑥+1 2 = 𝐴 𝑥−1 + 𝐵𝑥+𝐶 𝑥2+𝑥+1 + 𝐷𝑥+𝐸 𝑥2+𝑥+1 2 • 3𝑥−2 𝑥−1 2 𝑥2+𝑥+1 = 𝐴 𝑥−1 + 𝐵 𝑥−1 2 + 𝐶𝑥+𝐷 𝑥2+𝑥+1
- 42. Question: Find A andB such that 𝑥+10 𝑥2−4 = 𝐴 𝑥−2 + 𝐵 𝑥+2 . a. A= -3; B= 2 b. A= -3; B= -2 c. A= 3; B= -2 d. A= 3; B= 2 ANS: C 𝑥 + 10 = 𝐴 𝑥 + 2 + 𝐵 𝑥 − 2 𝑥 + 10 = 𝐴 + 𝐵 𝑥 + 2𝐴 − 2𝐵 This implies 𝐴 + 𝐵 = 1 and 2𝐴 − 2𝐵 = 10 Then, 2 1 − 𝐵 − 2𝐵 = 10 −4𝐵 = 8 𝑩 = −𝟐 From 𝐴 + 𝐵 = 1 𝑨 = 𝟑
- 43. Question: Find the valueof A: 𝑥2 + 4𝑥 + 10 𝑥3 + 2𝑥2 + 5𝑥 = 𝐴 𝑥 + 𝐵 2𝑥 + 2 𝑥2 + 2𝑥 + 5 + 𝐶 𝑥2 + 2𝑥 + 5 a. 2 b. -2 c. ½ d. -1/2 ANS: A 𝑥2 + 4𝑥 + 10 = 𝐴 𝑥2 + 2𝑥 + 5 + 𝐵𝑥 2𝑥 + 2 + 𝐶𝑥 𝑥2 + 4𝑥 + 10 = 𝐴 + 2𝐵 𝑥2 + 2𝐴 + 2𝐵 + 𝐶 𝑥 + 5𝐴 5𝐴 = 10 𝑨 = 𝟐
- 44. Progression DESCRIPTION LAST TERMSUM MEAN Arithmetic Progression Sequence of number with a fixed common difference i.e., 1, 3, 5, 7, 9 𝐿 = 𝑎 + 𝑛 − 1 𝑑 𝑆 = 𝑛 2 (𝑎 + 𝐿) 𝑏 = 𝑎 + 𝑐 2 Geometric Progression Sequence of number with a fixed common ratio i.e., 1, 3, 9, 27 𝐿 = 𝑎𝑟𝑛−1 𝑆 = 𝑎 1 − 𝑟𝑛 1 − 𝑟 𝑏 = ± 𝑎𝑐 Harmonic Progression The sequence of numbers formed by the reciprocals of the terms of an arithmetic progression (i.e., 1, 1 3 , 1 5 , 1 7 , 1 9 ).
- 45. Question: What is the30th element of the arithmetic sequence for which the first element is 5 and the third is 13? a. 237 b. 125 c. 121 d. 150 ANS: C Given: 𝑎 = 5; 𝑎3 = 13; 𝑛 = 30 𝑎2 = 5 + 13 2 = 9 𝑑 = 4 𝑎30 = 5 + 30 − 1 4 𝒂𝟑𝟎 = 𝟏𝟐𝟏
- 46. Question: Find the 12thterm of the series 6, 3, 2. a. ½ b. ¼ c. 1/8 d. 1/12 ANS: A 𝑁𝑜𝑡𝑒: 1 6 , 1 3 , 1 2 is an arithmetic seq. Given: 𝑎1 = 1 6 ; 𝑑 = 1 6 ; 𝑛 = 12 𝑎12 = 1 6 + 12 − 1 1 6 𝑎12 = 12 6 𝑜𝑟 2 Thus, the 12th term is 𝟏 𝟐 .
- 47. Question: The 1st termof a geometric progression is 64, the last term is -2 and the sum of the terms is 42. How many terms are there? a. 12 b. 10 c. 8 d. 6 ANS: D 42 = 64 1−𝑟𝑛 1−𝑟 −2 = 64𝑟𝑛−1 42 − 42𝑟 = 64 − 64𝑟𝑛 − 2𝑟 64 = 𝑟𝑛 42 − 42𝑟 = 64 − 64 − 2𝑟 64 42 − 42𝑟 = 64 + 2𝑟 − 1 32 = − 1 2 𝑛−1 44𝑟 = −22 − 1 2 5 = − 1 2 𝑛−1 𝑟 = − 1 2 𝑛 − 1 = 5 𝑜𝑟 𝑛 = 6
- 48. Problem Solving Step 1.Identify the given and unknowns Step 2. Translate word phrases into algebraic symbols Step 3. Perform the operations and solve for the unknowns Step 4. Verify your answer.
- 49. Question: One pipe canfill a tank in 45 minutes and another pipe can fill it in 30 minutes. If these two pipes are open while a 3rd pipe is drawing water from the tank, it takes 27 minutes to fill the tank. How long will it take the 3rd pipe alone to empty a full tank? a. 48 min b. 20 min c. 54 min d. 60 min ANS: C TIME RATE Pipe 1 45min 1/45 Pipe 2 30min 1/30 Pipe 3 X min 1/x 1 45 + 1 30 − 1 𝑥 = 1 27 𝑥 = 54 𝑚𝑖𝑛
- 50. Question: A student ina chemistry laboratory wants to form a 32 ml mixture of 2 solutions to contain 30% acid. Solution A contains 42% acid and solution B contains18% acid. How many ml of each solution must be used? a. A=16; B=16 b. A=14; B=18 c. A=10; B=22 d. A=15; B=17 ANS: A mL % Soln 1 X 42% Soln 2 y 18% 𝑥 + 𝑦 = 32 0.42𝑥 + 0.18𝑦 = 32 0.3 0.42 32 − 𝑦 + 0.18𝑦 = 9.6 13.44 − 0.24𝑦 = 9.6 𝑦 = 16 𝑥 = 16
- 51. Question: Mary was fourtimes as old as Ann four years ago and if Mary will be twice as old as Ann four years hence. How old is Ann? a. 14 b. 12 c. 10 d. 8 ANS: D -4 Present +4 Mary 4(x-4) 4x-8 Ann X-4 x X+4 2 𝑥 + 4 = 4𝑥 − 8 2𝑥 = 16 𝑥 = 8
- 52. Question: The number ofcentimeters in the perimeter of a certain square is equal to the number of square centimeter in its area. Find the length of the sides of the square. a. 5 cm b. 2 cm c. 4 cm d. 6 cm ANS: C 4𝑠 = 𝑠2 𝑠2 − 4𝑠 = 0 𝑠 𝑠 − 4 = 0 𝑠 = 0 𝑜𝑟 𝒔 = 𝟒
- 53. Question: Find two consecutiveeven integers such that the square of larger is 44 greater than the square of the smaller integer. a. 10 & 12 b. 12 & 14 c. 8 & 10 d. 14 & 16 ANS: A Let 𝑥 and 𝑥 + 2 be the two even integers 𝑥 + 2 2 = 𝑥2 + 44 𝑥2 + 4𝑥 + 4 = 𝑥2 + 44 4𝑥 = 40 𝒙 = 𝟏𝟎 𝒙 + 𝟐 = 𝟏𝟐
- 54. Question: A boat travelsat the rate of 28 kph in still water. It took the boat 2.75 hrs to travel downstream and 4.25 hrs to cover the same distance upstream. Find the rate of the water current. a. 5 kph b. 6 kph c. 7 kph d. 8 kph ANS: B 𝐷 = 𝑅𝑇 2.75 𝑥 + 28 = 4.25 28 − 𝑥 2.75𝑥 + 77 = −4.25𝑥 + 119 7𝑥 = 42 𝑥 = 6
- 55. Question: The unit digitsof a 2-digit no. exceeds the tens digit by 3. If the digits are reversed and divided by the original, the quotient is 2 and remainder is 2. Find the number. a. 25 b. 36 c. 14 d. 22 ANS: A 𝑑𝑖𝑔𝑖𝑡 = 10𝑥 + 𝑥 + 3 Reverse = 10 𝑥 + 3 + 𝑥 10 𝑥 + 3 + 𝑥 10𝑥 + 𝑥 + 3 = 2 + 2 10𝑥 + 𝑥 + 3 11𝑥 + 30 11𝑥 + 3 = 22𝑥 + 8 11𝑥 + 3 11𝑥 + 30 = 22𝑥 + 8 11𝑥 = 22 𝑥 = 2 Digit = 20 + 2 + 3 = 25
- 56. Complex Number: Complex numbersare written in the form 𝑎 + 𝑏𝑖, where 𝑎 is the real part and 𝑏𝑖 is the imaginary part. When dealing with imaginary number always remember that 𝑖2 = −1. Example: 𝑖15 = 𝑖14 𝑖 = 𝑖2 7 𝑖 = −1 7 𝑖 = −𝑖
- 57. Question: (3 – 2i)(4+ 2i) is equal to ________. a. 12 – 4i b. 8 – 2i c. 16 – 2i d. 8 + 2i ANS: C = 12 − 8𝑖 + 6𝑖 − 4𝑖2 = 12 − 2𝑖 − 4 −1 = 16 − 2𝑖