MATH 150 Pre-Calculus

Transcription

1 MATH 150 Pre-Calculus Fall, 2014, WEEK 2 JoungDong Kim Week 2: 1D, 1E, 2A Chapter 1D. Rational Expression. Definition of a Rational Expression A rational expression is an expression of the form p, where p and q are polynomials, and q cannot q be the zero polynomial. Note. Restrictions occur whenever the denominator is zero. Ex1) Find a domain. a) 4x 7 6x 16 1

2 b) 2 x 9 c) x+5 x d) x+1 x 2 7x+10 2

3 Simplifying Rational Expressions To simplifying a rational expression means to reduce it to lowest terms by cancelling common factors. Therefore, the key to simplifying is to factor the polynomials. Ex2) a) 8x4 12x 6 b) x3 +5x 2 +6x x 2 4 c) 4x x 3 x 2 x 2 3

4 Operations with Rational Expressions 1. Multiplication of Rational Expressions. a b c d = ac bd. Also, when multiplying fractions that you may cross-cancel. Ex3) a) b) 6x3 5x 10x 21 c) 4 x 1 x x 32 d) x2 x 12 9x 2 6x 3x3 2x 2 x+3 e) 2x2 +x 6 x 2 +4x 5 x3 3x 2 +2x x+5 4x 2 6x x 3 8 4

5 2. Division of Rational Expressions. a b c d = a b d c = ad bc Ex4) a) x2 +2x 15 x+2 (x 2 +7x+10) 5

6 b) y2 7y +12 y 2 +3y 18 y2 +3y 28 3y +21 y 2 +12y+36 4y +24 Note. p(x) q(x), q, r, and s 0. r(x) s(x) 6

7 c) x2 +3x 10 x 2 +6x+5 x2 +2x+1 2x 2 x 6 2x+2 2x+3 7

8 3. Addition and Subtraction of Rational Expressions. Find a common denominator. The least common denominator (LCD) is the product of all prime factors of the denominator. Ex5) a) 1 x 2 x 1 b) y 4 y +6 y2 +3y 28 y 2 +12y +36 8

9 c) x 5 (x 7)(x+1) 1 x 2 4 d) 7 2x+1 8x 2x

10 Compound Fractions A compound Fraction (or Complex fraction) is an expression which contains fractions within fractions. To simplify, Step 1) Simplify both the numerator and denominator individually. Step 2) Divide the numerator by the denominator by multiplying with the reciprocal of the denominator. 10

11 Ex6) 6 y 5 2y +1 6 y +4 11

12 Ex7) Simplify the expression 2(x+h+1) 1 2(x+1) 1 h (Common one in many calculus calsses.) 12

13 Ex8) If f(x) = x 2 +x, find and simplify f(x) f(2). x 2 13

14 Chapter 1E. Complex Numbers. Definition of Complex Number A complex number is a number that can be written in the form a+bi, where a, b are real numbers and i = 1. a is referred to as the real part and b is referred to as the imaginary part. The standard form of the complex number is a+bi. 14

15 Ex9) Find real numbers a and b such that ( ) b a) (2a 1) 4i = 7+ i 3 b) (a+b)+(a b)i = 1 15

16 Absolute value of a complex number If z = a+bi is a complex number, z = a+bi = a 2 +b 2. The distance between two complex number z 1 and z 2 is z 1 z 2. Ex10) Compute the absolute values of the following complex numbers. a) 2 5i b) 2+ 9 c) 5 d) 3i 16

17 Properties of the absolute value of complex number 1. For z = a+bi a complex number, the absolute value of z represents the distance from the origin (0,0) to the point with coordinate (a,b). 2. Let z 1 and z 2 be any two complex numbers then z 1 z 2 = z 1 z 2 and z 1 = z 1 z 2 3. Let z 1 and z 2 be any two complex numbers then z 2 z 1 +z 2 z 1 + z 2 This inequality is called the triangle inequality. Ex11) Show that the absolute value of the sum of 3+i and 2+2i is less than or equal to the sum of their absolute values. 17

18 Conjugate of a complex number The conjugate of the complex number z = a+bi is defined to be a bi denoted z. Ex12) Compute the conjugate of the following complex numbers. a) 1 2i b) 4 c) 4i d) 2+4i 18

19 Properties of the conjugate of a complex number 1. z z = z 2 2. If x is real number, then x = x. 3. z = z 4. z 1 +z 2 = z 1 +z 2 5. z 1 z 2 = z 1 z 2 6. ( z1 z 2 ) = z 1 z 2 19

20 Ex13) Express the following in the form a+bi. a) 7+3i 4i b) 3+5i 1 2i 20

21 Chapter 2. Equations and Inequalities. Chapter 2A. Solving Equations. The values of the unknown that make the equation true are called the solutions or roots of the equation, and the process of finding the solutions is called solving the equation. Solving Linear Equations A linear equation is an equation equivalent to one of the form Linear equation is solved by isolating the variable. Ex14) Solve the equation for x. a) 7x 4 = 3x+8 ax+b = 0. b) 6+3x = x 4 Ex15) Solve for the variable M in the equation, F = G mm r 2. 21

22 Solving Quadratic Equations A quadratic equation is an equation that can be written in the form 1. Solving by Factoring ax 2 +bx+c = 0. Theorem 0.1 (Zero-Product Property). A B = 0 if and only if A = 0 or B = 0. This means that if we can factor the left-hand side of a quadratic equation then we can solve it by setting each factor equal to 0 in turn. Note. This method works only when the right-hand side of the equation is 0. Ex16) Solve the equations a) x 2 +5x = 24 b) 2a 2 +3a = 14 c) x 3 x 2 6x = 0 22

23 2. Solving by Completing the Square. The solutions of the equation x 2 = c are x = c and x = c. Ex17) Solve each equation a) x 2 = 5 b) (x 4) 2 = 5 As we saw in the Ex17, if a quadratic equation is of the form (x±a) 2 = c, then we can solve it by taking the square root of each side. In an equation of this form the left-hand side is a perfect square. So if a quadratic equation does not factor readily, then we can solve it using the technique of Completing the Square. 23

24 Completing the Square To make x 2 + bx a perfect square, add the perfect square x 2 +bx+ ( ) 2 b, the square of half the coefficient of x. This gives 2 ( ) 2 ( b = x+ b ) Ex18) Solve the equation by Completing the Square. a) x 2 8x+13 = 0 24

25 b) 3x 2 12x+6 = 0 25

26 c) 9x 2 +10x+1 = 0 26

27 3. Solving with the Quadratic Formula. Completing the square on the general quadratic equation ax 2 +bx+c = 0. 27

28 Ex19) Solve the equation using the Quadratic formula. a) x 2 +7x+3 = 0 b) 3x 2 5x 1 = 0 28

29 Ex20) Solve the equation. a) x 2 6x 7 = 0 29

30 b) 3x 2 4x 15 = 0 30

31 Equations in Quadratic Form ax 2n +bx n +c = 0 If we define u = x n, then u 2 = (x n ) 2 = x 2n. Therefore, our equation above becomes au 2 +bu+c = 0 which is Quadratic form. Ex21) Solve b 4 9b = 0. Ex22) Solve x 2 3 4x 1 3 = 3. 31

32 Rational Equations The easiest way to solve a rational equation is to eliminate the fractions by multiplying both sides of the equation by the least common denominator (LCD). Ex23) Solve 2 x 3 x 2 = 1. 32

33 Ex24) Solve 2+ 5 x 4 = x+1 x 4. 33

34 Radical Equations Step 1. Isolate one radical. Step 2. Raise both sides of the equation to the appropriate power to remove the radical. Step 3. If necessary, repeat the process until all radicals have been removed, then solve the resulting equation. Step 4. Check! your solutions are in the domain. Ex25) Solve 3 x+7 4 = 8. 34

35 Ex26) Solve x+5+1 = x. 35

36 Ex27) Solve 2 x x 3 = 3. 36

37 Absolute Value Equations Recall that x if x 0 x = x if x < 0 To solve the absolute value equations, we rewrite every absolute value equation as two separate equations. Ex28) Solve x 7 = 3. Recall that a b refers to the distance between the point a and b on the number line. Therefore, the equation x 7 = 3 can be thought of as, the distance between x and 7 is 3? or what number or numbers are 3 units from 7 on the number line? 37

38 Ex29) Solve the equation x(x 1) = 2. 38

39 Equations in Several Variables Ex30) Solve S = 2πr 2 +2πrh for h. Ex31) Solve S = 2πr 2 +2πrh for r. 39

40 Ex32) Suppose an object is dropped from a height h 0 above the ground. Then its height after t seconds is given by h = 16t 2 +h 0, where h is measured in feet. If a ball is dropped from 288 ft above the ground, how long does it take to reach ground level? 40