Discrete Mathematics

Answer Key

1. Simplify the following expression: (2^{2}x10^{3})^{2} **16,000,000**

2. Find the product of (4x-5)(2x+3) **8x ^{2} + 2x -15.**

3. Factor the following trinomial: 6x^{2} - 13x + 6 **(3x-2)(2x-3)**

4. Use the quadratic formula to solve: 2x^{2} + 7x - 5 = 0 **x = (-7 +/- sqrt(89))/4**

5. Find all solutions to the following equation: x^{4} - 11x^{2} + 28
**x = 2, -2, sqrt(7), -sqrt(7)**

6. Find all solutions to the following equation: | 3x - 1 | = 7 **x = -2, 8/3**

7. Solve the inequality and sketch the solution on the real number line: 4 - 2x < 12**
x > -4 (graph missing)**

8. Plot the points and find the slope of the line passing through (-3, -1) and (1, 7). **slope
= 2 (graph missing)**

9. Sketch the graph of the quadratic function. Identify the vertex and intercepts: f(x)=x^{2}
- 11x + 28

**x-intercept: (4, 0) and(7, 0). y-intercept: (0, 28). Vertex: (11/2, 37/4). (graph missing)**

10. Divide by long division: (4x^{3} - 8x^{2} - 11x + 5) / ( x + 4)
**4x ^{2} - 24x + 85 - 335/(x+4)**

11. Divide by synthetic division: (3x^{3} - 17x^{2} + 15x - 25)/(x-5)
**3x ^{2} - 2x + 5**

12. Evaluate the following expression: log_{2}(4^{7}). **14**

13. Write the following expression as the logarithm of a single expression: 2log_{5}x
+ 3log_{5}y - 5log_{5}z

**log _{5}(x^{2}y^{3}/z^{5})**

14. Solve the following system of equations: 3x - 5y = 13, 2x + 5y = 7. **x= 4, y = -1/5**

15. Solve the following system of equations: 6x - 5y = 1, 4x + 7y = 11. **x =1, y=1**

16. Find the midpoint of the interval with endpoints -2 and 15. **13/2**

17. Suppose in the interval [a,b], a = -2. Find b if the midpoint of the interval is 7.
**b = 16**

18. Find the point that is five-ninths of the way from -4 to 23. **11**

19. If 4 is midway between a and b and 8 is two-thirds of the way from a to b, what are a and
b?

** a = -8, b = 16**

20. What is the set of letters in the state name MISSISSIPPI? How many elements are in that
set?

**{ M, I, S, P }, 4**

21. How many elements are in the set {1, 2, {3, 4}, {5, 6, 7} }? ** 4**

How many elements are in the set {1,2,3,4,5,6,7}? **7**

22. What is the power set of {a, b}? **{ {}, {a}, {b}, {a,b} }**

23. What is the power set of the empty set? **{ {} }**

What is the power set of { {} }? **{ {}, {{}} }**

24. Let A = {a, b, c, d, e} and B = {c, d, e, f, g}.

What is A intersect B? {c, d, e}

What is A union B? {a, b, c, d, e, f, g}

What is A - B? {a, b}

25. Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}?

How many elements are in 2^{A}? **2 ^{12}**

How many elements in 2

How many elements in 2