# Find the greatest common factor. 4, 6, 12.

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**Directions**: Please show all of your work for each problem. If applicable, you may find Microsoft Word’s equation editor helpful in creating mathematical expressions in Word. There is a tutorial on using this equation editor in Module 1 Lecture Notes. You also have the option of hand writing your work and scanning it.

1. Find the greatest common factor. 4, 6, 12.

2. Factor. 24*x*^{3} + 30*x*^{2}

3. Factor out the GCF with a negative coefficient. –24*m*^{2}*n*^{6} – 8*mn*^{5} – 32*n*^{4}

4. Factor completely by factoring out any common factors and then factoring by grouping.

6*x*^{2 }– 5*xy* + 6*x* – 5*y*

5. The GCF of 15*y* + 20 is 5. The GCF of 15*y* + 21 is 3. Find the GCF of the product (15*y* + 20)(15*y* + 21).

6. The area of a rectangle of length *x* is given by 15*x* – *x*^{2}. Find the width of the rectangle in terms of *x*.

7. Factor the trinomial completely. *x*^{2} + 8*x* – 9

8. Factor the trinomial completely. 2*x*^{2} + 16*x* + 32

9. Complete the following statement. 6*a*^{2} – 5*a* + 1 = (3*a* – 1)(__?__)

10. State whether the following is true or false. *x*^{2} – 7*x* – 30 = (*x* + 3)(*x *– 10)

11. Factor completely. *x*^{2} + 11*x* + 28

12. Factor completely. 15*x*^{2} + 23*x* + 4

13. Factor completely. 6*z*^{3} – 27*z*^{2} + 12*z*

14. The number of hot dogs sold at the concession stand during each hour ii*h* after opening at a soccer tournament is given by the polynomial 2*h*^{2} – 19*h* + 24. Write this polynomial in factored form.

15. Find a positive value for *k* for which the polynomial can be factored.*x*^{2} – *kx* + 29

16. Factor completely. 9*x*^{2} + 4

17. Determine whether the following trinomial is a perfect square. If it is, factor the binomial.*x*^{2} – 12*x* + 36

18. Factor completely. 25*x*^{2} + 40*xy* + 16*y*^{2}

19. Factor. *s*^{2}(*t *– *u*) – 9*t*^{2}(*t* – *u*)

20. State which method should be applied as the first step for factoring the polynomial. 6*x*^{3} + 9*x*

21. State which method should be applied as the first step for factoring the polynomial. 2*a*^{2} + 9*a* + 10

22. Solve the quadratic equation. 5*x*^{2} + 17*x =* –6

23. Solve the quadratic equation. 3*x*(2*x* – 15) = –84

24. The sum of an integer and its square is 30. Find the integer.

25. If the sides of a square are decreased by 3 cm, the area is decreased by 81 cm^{2}. What were the dimensions of the original square?

26. Write in simplest form.

27. Write in simplest form.

28. Write the expression in simplest form.

29. The area of the rectangle is represented by 5*x*^{2} + 19*x* + 12. What is the length?

5*x* + 4

30. Multiply.

31. Multiply.

32. Divide.

33. Divide.

34. Perform the indicated operations.

35. Find the area of the rectangle shown.

36. Subtract. Express your answer in simplest form.

37. Subtract. Express your answer in simplest form.

38. Add. Express your answer in simplest form.

39. Add. Express your answer in simplest form.

40. Add or subtract as indicated.

41. One number is 8 less than another. Let *x* represent the larger number and use a rational expression to represent the sum of the reciprocals of the two numbers.

42. Simplify.

43. Simplify.

44. What values for *x*, if any, must be excluded in the following algebraic fraction?

45. What values for *x*, if any, must be excluded in the following algebraic fraction?

46. Solve for *x*. + 6 = 1

47. Solve for *x*.

48. Solve for *x*.

49. One number is 3 times another. If the sum of their reciprocals is , find the two numbers.

50. A 5-foot pole casts a shadow of 4 feet. How tall is a tree with a shadow of 16 feet?

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