ASSIGNMENT 1 - MATH 112

QUESTIONS ARE BASED ON MATERIAL FROM THE PREREQUISITES CHAPTER, SECTIONS 1 THROUGH 6,

OF YOUR TEXTBOOK. (SAME SECTIONS IF YOU ARE USING THE SECOND EDITION OF THE TEXTBOOK.)

(1) Rewrite using set-builder notation each set that is described below in words.

(a) A is the set of natural numbers which are greater than or equal to 4 and strictly less than 79.

(b) B is the set of rational numbers which are less than or equal to 125.

(c) C is the set of real numbers lying strictly between -73 and 58.

(2) Let A,B and C be the sets defined above. Write the following in set-builder notation.

(a) A C(b) B A

(3) Express the given intervals using inequalities.

(a) (4,)(b) [3, 9)

(4) Express the given inequalities using interval notation.

(a) x > 1(b) 10 < x 8

(5) Evaluate each expression.

(a) 3125

(b)28

(c) 642/3

(6) Write each expression as a power of x.

(a) 1x3

(b) (x2xm)n

(c) (x)3xa

xb

(7) Simplify each expression.

(a)x2y4/3

x1/3y

6(b)

ab2c32a3b4

2(c) |2 |2 | 2|||

1

(8) Write the following numbers in scientific notation.

(a) 6,403,800,000,000,000

(b) 0.0000000000007513

(9) Express the decimal 0.246 as a fraction.

(10) Factor each expression completely.

(a) y3 2y2 y + 2(b) x4 2x2 + 1(c) 2x2 + 7x 4(d) 8x3 + y6

(e) a4b2 + ab5

(11) If t = 12x3 1x3

and x > 0, show that

1 + t2 =

1

2

x3 +

1

x3

.

Why is it the requirement x > 0 necessary in order for the final statement to be true?

(12) This problem demonstrates one of the most powerful techniques used in mathematics: noticing

a pattern, and using that pattern to make a more general statement. Follow the steps below to

derive a factorization of the nth-degree polynomial xn 1.(a) Factor the expression x2 1.(b) Factor the expression x3 1.(c) Factor the expression x4 1.(d) Write down a factorization of xn 1, for any n N, of the following form:

xn 1 = (x 1)p

where p is an (n 1)-degree polynomial in x.Hint: It might not be fruitful to factor each expression in (a)-(c) completely. You only want to

factor out the term (x 1) in order to see the pattern.