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Theorem 7.1: If f (x) = an x n + an 1 x n 1 + + a1 x + a0 is a polynomial with integral coef cients (that is, the numbers an , an 1 , , a1 , a0 are integers), and if r is an integer that is a root of f , then r must be a divisor of the constant term a0 . EXAMPLE By Theorem 7.1, any integral root of x3 2x2 5x + 6 must be among the divisors of 6, which are 1, 2, 3, and 6. By actual substitution, it is found that 1, 2, and 3 are roots. Theorem 7.2: A number r is a root of the polynomial f (x) = an x n + an 1 x n 1 + + a1 x + a0 if and only if f (x) is divisible by the polynomial x r. For hints for the proofs of Theorems 7.1 and 7.2, see Supplementary Problems 7.26 and 7.27. EXAMPLES

1 15.29. Evaluate Ans. (a)

(a) Consider the polynomial f (x) = x 3 4x 2 + 14x 20. By Theorem 7.1, any integral root must be a divisor of 20; that is, 1, 2, 4, 5, 10, or 20. Calculation reveals that 2 is a root. Hence, by Theorem 7.2, x 2 must divide f (x); we carry out the division in Fig. 7-10(a). Since f (x) = (x 2)(x 2 2x + 10), its remaining roots are found by solving x 2 2x + 10 = 0

x6 e 3x dx;

Fig. 7-10

1 c

Note If your application uses an object of another application, you can t remove the reference to it

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7.24 Express the set of solutions of each inequality below in terms of the notation for intervals: (a) 2x + 3 < 9 (e) 3 < 4x 5 < 7 2x 5 <1 (i) x 2 (b) 5x + 1 6 (f ) 1 2x + 5 < 9 (j) x 2 6 (c) 3x + 4 5 (g) |x + 1| < 2 (k) (x 3)(x + 1) < 0 (d) 7x 2 > 8 (h) |3x 4| 5

x2 e 2x dx:

7.25 For what values of x are the graphs of (a) f (x) = (x 1)(x + 2) and (b) f (x) = x(x 1)(x + 2) above the x-axis GC Check your answers by means of a graphing calculator. 7.26 Prove Theorem 7.1. [Hint: Solve f (r) = 0 for a0 .] 7.27 Prove Theorem 7.2. [Hint: Make use of the algebra following Problem 7.21.]

a 24; 1

8.1 INTRODUCTION To a great extent, calculus is the study of the rates at which quantities change. It will be necessary to see how the value f (x) of a function f behaves as the argument x approaches a given number. This leads to the idea of limit. EXAMPLE Consider the function f such that

80 ; b 243 e x dx;

CHAPTER 1 THE VBA INTEGRATED DEVELOPMENT ENVIRONMENT (VBAIDE)

whenever this formula makes sense. Thus, f is de ned for all x for which the denominator x 3 is not 0; that is, for x = 3. What happens to the value f (x) as x approaches 3 Well, x 2 approaches 9, and so x 2 9 approaches 0; moreover, x 3 approaches 0. x2 9 Since the numerator and the denominator both approach 0, it is not clear what happens to . x 3 However, upon factoring the numerator, we observe that x2 9 (x 3)(x + 3) =x+3 = x 3 x 3 Since x + 3 unquestionably approaches 6 as x approaches 3, we now know that our function approaches 6 as x approaches 3. The traditional mathematical way of expressing this fact is x2 9 =6 x 3 x 3 lim This is read: The limit of x2 9 as x approaches 3 is 6. x 3 Notice that there is no problem when x approaches any number other than 3. For instance, when x approaches 4, x 2 9 approaches 7 and x 3 approaches 1. Hence, 7 x2 9 = =7 1 x 4 x 3 lim

0 1 3

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