Precalculus Review I Precalculus Review II The Cartesian Coordinate System Straight Lines

1. 1 • Precalculus Review I • Precalculus Review II • The Cartesian Coordinate System • Straight Lines Preliminaries

2. 1.1 Precalculus Review I

3. The Real Number Line • We can represent real numbers geometrically by points on a real number, or coordinate, line: • This line includes allreal numbers. • Exactly one point on the line is associated with each real number, and vice-versa. Origin Negative Direction Positive Direction – 4 – 3 – 2 – 1 0 1 2 3 4 p

4. Finite Intervals Open Intervals • The set of real numbers that lie strictlybetween two fixed numbersa and b is called an open interval(a, b). • It consists of all the real numbers that satisfy the inequalitiesa < x < b. • It is called “open” because neither of its endpoints is included in the interval.

5. Finite Intervals Closed Intervals • The set of real numbers that lie between two fixed numbersa and b, that includesa and b, is called a closed interval[a, b]. • It consists of all the real numbers that satisfy the inequalitiesa  x  b. • It is called “closed” because both of its endpoints are included in the interval.

6. Finite Intervals Half-Open Intervals • The set of real numbers that between two fixed numbersa and b, that contains only one of its endpointsa or b, is called a half-open interval(a, b] or [a, b). • It consists of all the real numbers that satisfy the inequalitiesa < x  b or a  x < b.

7. Infinite Intervals Examples of infinite intervals include: • (a, ), [a, ), (–, a), and (–, a]. • The above are defined, respectively, by the set of real numbers that satisfy x > a, x  a, x < a, x  a.

8. Exponents and Radicals • If b is any real number and n is a positive integer, then the expression bn is defined as the number bn = b · b b ·· … · b • The number b is called the base, and the superscript n is called the power of the exponential expressionbn. • For example: • If b≠ 0, we define b0 = 1. • For example: 20 = 1 and (–p)0 = 1, but 00 is undefined. nfactors

9. Exponents and Radicals • If n is a positive integer, then the expression b1/n is defined to be the number that, when raised to the nth power, is equal to b, thus • Such a number is called the nthroot ofb, also written as • Similarly, the expression bp/q is defined as the number (b1/q)p or • Examples: (b1/n)n = b

10. Laws of Exponents Law Example 1. am· an = am + n x2· x3 = x2 + 3 = x5 2. 3. (am)n = am · n (x4)3 = x4 · 3 = x12 4. (ab)n = an · bn (2x)4 = 24 ·x4= 16x4 5.

11. Examples • Simplify the expressions Example 1, page 6

12. Examples (assume x, y, m, and n are positive) • Simplify the expressions Example 2, page 7

13. Examples • Rationalize the denominator of the expression Example 3, page 7

14. Examples • Rationalize the numerator of the expression Example 4, page 7

15. Operations With Algebraic Expressions • An algebraic expression of the form axn, where the coefficient a is a real number and n is a nonnegative integer, is called a monomial, meaning it consists of one term. Examples: 7x22xy12x3y4 • A polynomial is a monomial or the sum of two or more monomials. Examples: x2 + 4x + 4 x4 + 3x2 – 3 x2y – xy + y

16. Operations With Algebraic Expressions • Constant terms, or terms containing the same variable factors are called like, or similar, terms. • Like terms may be combined by adding or subtracting their numerical coefficients. • Examples: 3x + 7x = 10x 12xy – 7xy = 5xy

17. Examples • Simplify the expression Example 5, page 8

18. Examples • Simplify the expression Example 5, page 8

19. Examples • Perform the operationand simplify the expression Example 6, page 8

20. Examples • Perform the operationand simplify the expression Example 6, page 9

21. Factoring • Factoring is the process of expressing an algebraic expression as a product of other algebraic expressions. • Example:

22. Factoring • To factor an algebraic expression, first check to see if it contains any common terms. • If so, factor out the greatest common term. • For example, the greatest common factor for the expression is 2a, because

23. Examples • Factor out thegreatest common factorin each expression Example 7, page 9

24. Examples • Factor out thegreatest common factorin each expression Example 8, page 10

25. Factoring Second Degree Polynomials • The factors of the second-degree polynomial with integral coefficients px2 + qx + r are (ax + b)(cx + d), where ac = p, ad + bc = q, and bd = r. • Since only a limited number of choices are possible, we use a trial-and-error method to factor polynomials having this form.

26. Examples • Find the correct factorization for x2 – 2x – 3 Solution • The x2coefficient is 1, so the only possible first degree terms are (x )(x ) • The product of the constant term is –3, which gives us the following possible factors (x – 1)(x + 3) (x + 1)(x – 3) • We check to see which set of factors yields –2 for the xcoefficient: (–1)(1) + (1)(3) = 2 or (1)(1) + (1)(–3) = –2 and conclude that the correct factorization is x2 – 2x – 3 = (x +1)(x –3)

27. Examples • Find the correct factorization for the expressions Example 9, page 11

28. Roots of Polynomial Expressions • A polynomial equation of degree n in the variable x is an equation of the form where n is a nonnegative integer and a0, a1, … , an are real numbers with an≠ 0. • For example, the equation is a polynomial equation of degree 5.

29. Roots of Polynomial Expressions • The roots of a polynomial equation are the values of x that satisfy the equation. • One way to factor the roots of a polynomial equation is to factor the polynomial and then solve the equation. • For example, the polynomial equation may be written in the form • For the product to be zero, at least one of the factors must be zero, therefore, we have x = 0 x – 1 = 0 x – 2 = 0 • So, the roots of the equation are x = 0, 1, and 2.

30. The Quadratic Formula • The solutions of the equation ax2 + bx + c = 0 (a≠ 0) are given by

31. Examples • Solve the equation using the quadratic formula: Solution • For this equation, a = 2, b = 5, and c = –12, so Example 10, page 12

32. Examples • Solve the equation using the quadratic formula: Solution • First, rewrite the equation in the standard form • For this equation, a = 1, b = 3, and c = –8, so Example 10, page 12

33. 1.2 Precalculus Review II

34. Rational Expressions • Quotients of polynomials are called rational expressions. • For example

35. Rational Expressions • The properties of real numbers apply to rational expressions. Examples • Using the properties of number we may write where a, b, and c are any real numbers and b and c are not zero. • Similarly, we may write

36. Examples • Simplify the expression Example 1, page 16

37. Examples • Simplify the expression Example 1, page 16

38. Rules of Multiplication and Division If P, Q, R, and S are polynomials, then • Multiplication • Division

39. Example • Perform the indicated operation and simplify Example 2, page 16

40. Rules of Addition and Subtraction If P, Q, R, and S are polynomials, then • Addition • Subtraction

41. Example • Perform the indicated operation and simplify Example 3b, page 17

42. Other Algebraic Fractions • The techniques used to simplify rational expressions may also be used to simplify algebraic fractions in which the numerator and denominator are not polynomials.

43. Examples • Simplify Example 4a, page 18

44. Examples • Simplify Example 4b, page 18

45. Rationalizing Algebraic Fractions • When the denominator of an algebraic fraction contains sums or differences involving radicals, we may rationalize the denominator. • To do so we make use of the fact that

46. Examples • Rationalize the denominator Example 6, page 19

47. Examples • Rationalize the numerator Example 7, page 19

48. Properties of Inequalities • If a, b, and c, are any real numbers, then Property 1If a<b and b < c, then a < c. Property 2 If a<b, then a + c < b + c. Property 3 If a<b and c > 0, then ac < bc. Property 4 If a<b and c < 0, then ac > bc.

49. Examples • Find the set of real numbers that satisfy –1  2x – 5 < 7 Solution • Add 5 to each member of the given double inequality 4  2x < 12 • Multiply each member of the inequality by ½ 2 x < 6 • So, the solution is the set of all values of x lying in the interval [2, 6). Example 8, page 20

50. Examples • Solve the inequality x2 + 2x – 8 < 0. Solution • Factorizing we get (x + 4)(x – 2) < 0. • For the product to be negative, the factors must have opposite signs, so we have two possibilities to consider: • The inequality holds if (x + 4) < 0 and(x – 2) > 0, which meansx < –4, andx > 2, but this is impossible: x cannot meet these two conditionssimultaneously. • The inequality also holds if (x + 4) > 0 and(x – 2) < 0, which meansx > –4, andx < 2, or –4 < x < 2. • So, the solution is the set of all values of x lying in the interval (–4, 2). Example 9, page 20