Module 01:Fundamentals of Algebra

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The KEY for this module is for you to study the functions of your CALCULATOR. Solving problems in Algebra is EASY when you know how to use your calculator. Take time to study the owner’s manual or clicking the link provided below.

SHORTCUT METHOD: You can actually use the choices and substitute it to the variables in the problem to determine the final answer or you also can assign numeric values to the variables and compare it with the given equation. 

  • Handouts
  • Exercises

LINK: Fundamentals of Algebra

Topics:

  • Properties of Algebra
  • Laws of Exponents
  • Laws of Radicals: Fractional Exponents
  • The Concept of Infinity
  • Indeterminate Forms
  • Factoring Techniques
  • Complex Numbers
  • Rules of Inequality
  • Functions
  • Determinants
  1. (∞ – 2800)/∞ = ?
    a. Indeterminate
    b. Infinity
    c. Undefined
    d. Plane
  2. y to the power of 1/∞ is equal to:
    a. 1
    b. zero
    c. Infinity
    d. Indeterminate
  3. Factor the expression as completely as possible: x2 + 6x + 9
    a. (x + 3)2
    b. (x + 1)2
    c. (x + 4)2
    d. (x – 3)2
  4. Factor the expression as completely as possible: x4 – y4
    a. (x2 + y2)(x – y)
    b. (x2 + y2)(x + y)
    c. (x2 + y2)(x – y)2
    d. (x2 + y2)(x – y)(x + y)
  5. Simplify: 3a(6b)0.5 + (6a2b)0.5 – (24a2b)0.5
    a. 2(6b)0.5
    b. 3a(6b)0.5
    c. 2a(6b)0.5
    d. 4a(6b)0.5
  6. Factor the expression as completely as possible: 4×3 – 4×2 – 24x
    a. x(x+3)(x-2)
    b. 4x(x+3)(x-2)
    c. x2(x+3)(x-2)
    d. 4x(x+3)(x+2)
  7. Solve for x: x = [(y2-4y+16)(y2-16)]/y3+64
    a. y-3
    b. y-2
    c. y-4
    d. y-5
  8. Simplify: 8b+2 – 9(8)b+1 + 64(8)b-2
    a. 8b
    b. 8b+1
    c. 8b-1
    d. -8b
  9. Solve for y: 2x – 3y = 10 and 3x + 2y = 6
    a. -18/13
    b. 18/13
    c. -13/18
    d. 13/18
  10. Simplify: 4(50y)0.5 – 4(8y)0.5
    a. 10(2y)0.5
    b. 12(2y)0.5
    c. 8(2y)0.5
    d. 14(2y)0.5
  11. Solve the equation simultaneously: x+y=-6 ; x+z=1 ; 2z-3y+2x=5
    a. (-5, -1, 6)
    b. (-5, -1, 8)
    c. (5, -1, -6)
    d. (-5, 0, 6)
  12. Solve the equation simultaneously: 5x+3y=6 and 6x-4y=-3
    a. 13/18, 51/38
    b. -15/38, 51/38
    c. -15/38, 51/38
    d. 15/38, 51/38
  13. Factor the expression completely: ax + ay – yb – bx.
    a. (x-y)(a+b)
    b. (x+y)(a+b)
    c. (x+y)(a-b)
    d. (x-y)(a-b)
  14. Factor the expression completely: 27 + x3
    a. (x+3)(x2-4x+10)
    b. (x+3)(x2+3x+9)
    c. (x+3)(x2-3x+9)
    d. (x+3)(x2+4x-10)
  15. i23 is equivalent to __.
    a. i
    b. –i
    c. 1
    d. 0
  16. Solve for x: 1/x + 1/(x+3) = 1
    a. -3
    b. 3
    c. 1
    d. -2
  17. Solve for y: y4 – 9y2 = 0
    a. -3,0,-3
    b. 3,-1,-3
    c. 0,1,3
    d. 0,-3,3
  18. Solve for y: y2-5y+6=0
    a. -2,3
    b. 3,-1
    c. 2,-3
    d. 3,2
  19. Solve for x: |6x + 18| = 4
    a. 7/3, 11/3
    b. -7/3, 11/3
    c. -7/3, -11/3
    d. 7/3, -11/3
  20. Solve the inequality s – 3 < 1 or 0.5s ≥ 3
    a. (s<4 or s≥6)
    b. (s<4 or s≤6)
    c. (s>4 or s≥6)
    d. (s≥4 or s≤6)
  21. Solve the inequality s – 8 < – 10 a. s<2 b. s>2
    c. s>-2
    d. s<-2
  22. For f(x) =x2+5x, find f(3x)
    a. 10×2-15x
    b. 9×2+15x
    c. 9×2-15x
    d. 9×2-20x
  23. Given the following vectors: A1 = 3i – 3j + 7k; A2 = 9i + 3j – 4k ; A3 = 2i – 5j – 2k solve for (A1+A2)*A3
    a. 20
    b. -18
    c. 18
    d. -25
  24. Find the dot product of B1 = 3i – 3j + 7k; B2 = 9i + 3j – 4k
    a. 8
    b. -9
    c. -10
    d. -8
  25. What is the determinant of A:
    2
    a. 70
    b. 75
    c. 80
    d. 85
  26. Find h(x) if h[f(x)] = (2×2-1)5 and f(x) = 2×2-1
    a. x5
    b. 2×2
    c. –x5
    d. -2×5
  27. Find the inverse of the function g(x) = 3x + 2
    a. (y-1)/2
    b. (y+3)/2
    c. (y+2)/3
    d. (y-2)3
  28. Let f(x)=(x-3)1/3 and h(x) = x3, find h[f(x)]
    a. x-2
    b. x-3
    c. x+3
    d. x-1
  29. Let f(x) = (x-3)1/3 and h(x) = x3 , find f[h(x)]
    a. (x2-2)1/3
    b. (x2+2)1/3
    c. (x2-3)1/3
    d. (x2+3)1/3
  30. For f(x) = (x-2)0.5, find f(3x)
    a. (5x-2)0.5
    b. (2x-1)0.5
    c. (3x+2)0.5
    d. (3x-2)0.5
  31. For f(x) = 3x – 2, find f(2)
    a. 4
    b. 3
    c. 5
    d. 2
  32. Find x: x = (7x – 10)0.5
    a. 5,2
    b. 5,-3
    c. 2,-5
    d. 2,4
  33. Solve the equation simultaneously: 2×2-3y2-14=0 and y=x-2
    a. (9,7), (3,1)
    b. (10,8), (-3,2)
    c. (9,-7),(3,2)
    d. (9,7),(-3,-1)
  34. Solve algebraically x and y: 3×2+8y2=35 and 12y2-4×2=42
    a. (1,2)
    b. (±1,±2)
    c. (-1,-2)
    d. (-1,2)
  35. Determine the value of m so that the equation 9×2 + mx + 1 = 0 , will just one real solution.
    a. ±6
    b. ±7
    c. ±5
    d. ±8
  36. Solve for y: y2-3y-10=0
    a. -2,-5
    b. -2,5
    c. 5,0
    d. -2,8
  37. Evaluate (3 + i)5
    a. 12+315i
    b. -12+300i
    c. -12+316i
    d. -10-310i
  38. Express 4 + 8i in polar form.
    a. 9.32 , 65.22o
    b. 10.32 , 60.22o
    c. 8.39 , 60.28o
    d. 8.94 , 63.43o
  39. What is the product of the complex numbers 5 + 8i and 7 – 3i?
    a. 59+41i
    b. 62+40i
    c. 50+35i
    d. 59+48i
  40. What is the exponential form of the complex number 1 + 5i?
    a. 5.1e78.7i
    b. 6.3e75.2i
    c. 4.3e70.2i
    d. 3.2e68.7i
  41. In the expansion of (x – 3y)7 find the y3-term, that is, the term involving y3.
    a. 949x4y3
    b. -940x4y3
    c. -945x4y3
    d. 900x4y3
  42. Find the 3rd term of the expansion of (3x – y)5.
    a. -270x3y2
    b. 270x3y2
    c. 270x2y2
    d. -270x2y2
  43. Find the sum of coefficients of (3x – 4y)12
    a. 4
    b. 3
    c. 2
    d. 1
  44. Find the constant term in the expansion of [3×3 – (1/x2)]5
    a. -90
    b. -92
    c. -91
    d. -89
  45. Resolve the following into partial fractions: (3×2 – 8x + 9)/(x-2)3
    a. [3/(x-2)] + [4/(x-2)2] + [5/(x-2)3]
    b. [3/(x+1)] – [2/(x+1)2] + [4/(x+1)3]
    c. [2/(x+1)] + [4/(x+1)2] + [3/(x+1)3]
    d. [4/(x+1)] – [2/(x+1)2] – [3/(x+1)3]

Answer Key (Handwritten)

    1. (∞ – 2800)/∞ = ?

    a. Indeterminate

    b. Infinity

    c. Undefined

    d. Plane

    2. y to the power of 1/∞ is equal to:

    a. 1

    b. zero

    c. Infinity

    d. Indeterminate

    3. Factor the expression as completely as possible: x2 + 6x + 9

    a. (x + 3)2

    b. (x + 1)2

    c. (x + 4)2

    d.  (x – 3)2

    4. Factor the expression as completely as possible: x4 – y4

    a. (x2 + y2)(x – y)

    b. (x2 + y2)(x + y)

    c. (x2 + y2)(x – y)2

    d. (x2 + y2)(x – y)(x + y)

    5. Simplify: 3a(6b)0.5 + (6a2b)0.5 – (24a2b)0.5 

    a. 2(6b)0.5

    b. 3a(6b)0.5

    c. 2a(6b)0.5

    d. 4a(6b)0.5

    6. Factor the expression as completely as possible: 4x3 – 4x2 – 24x

    a. x(x+3)(x-2)

    b. 4x(x+3)(x-2)

    c. x2(x+3)(x-2)

    d. 4x(x+3)(x+2)

    7. Solve for x: x = [(y2-4y+16)(y2-16)]/y3+64

    a. y-3

    b. y-2

    c. y-4

    d. y-5

    8. Simplify: 8b+2 – 9(8)b+1 + 64(8)b-2

    a. 8b

    b. 8b+1

    c. 8b-1

    d. -8b

    9. Solve for y: 2x – 3y = 10 and 3x + 2y = 6 

    a. -18/13

    b. 18/13

    c. -13/18

    d. 13/18

    10. Simplify: 4(50y)0.5 – 4(8y)0.5

    a. 10(2y)0.5

    b. 12(2y)0.5

    c. 8(2y)0.5

    d. 14(2y)0.5

    11. Solve the equation simultaneously: x+y=-6 ; x+z=1 ; 2z-3y+2x=5 

     a. (-5, -1, 6)

    b. (-5, -1, 8)

    c. (5, -1, -6)

    d. (-5, 0, 6)

    12. Solve the equation simultaneously: 5x+3y=6 and 6x-4y=-3 

    a. 13/18, 51/38

    b. -15/38, 51/38

    c. -15/38, 51/38

    d. 15/38, 51/38

    13. Factor the expression completely: ax + ay – yb – bx.

    a. (x-y)(a+b)

    b. (x+y)(a+b)

    c. (x+y)(a-b)

    d. (x-y)(a-b)

    14. Factor the expression completely: 27 + x3

    a. (x+3)(x2-4x+10)

    b. (x+3)(x2+3x+9)

    c. (x+3)(x2-3x+9)

    d.  (x+3)(x2+4x-10)

    15. i23 is equivalent to ____.

    a. i

    b. –i

    c. 1

    d. 0

    16. Solve for x: 1/x + 1/(x+3) = 1

    a. -3

    b. 3

    c. 1

    d. -2

    17. Solve for y: y4 – 9y2 = 0

    a. -3,0,-3

    b. 3,-1,-3

    c. 0,1,3

    d. 0,-3,3

    18. Solve for y: y2-5y+6=0

    a. -2,3

    b. 3,-1

    c. 2,-3

    d. 3,2

    19. Solve for x: |6x + 18| = 4

    a. 7/3, 11/3

    b. -7/3, 11/3

    c. -7/3, -11/3

    d. 7/3, -11/3

    20. Solve the inequality s – 3 < 1 or 0.5s ≥ 3

    a. (s<4 or s≥6)

    b. (s<4 or s≤6)

    c. (s>4 or s≥6)

    d. (s≥4 or s≤6)

    21. Solve the inequality s – 8 < – 10

    a. s<2

    b. s>2

    c. s>-2

    d. s<-2

    22. For f(x) =x2+5x, find f(3x)

    a. 10x2-15x

    b. 9x2+15x

    c. 9x2-15x

    d. 9x2-20x

    23. Given the following vectors: A1 = 3i – 3j + 7k; A2 = 9i + 3j – 4k ; A3 = 2i – 5j – 2k solve for (A1+A2)*A3 

    a. 20

    b. -18

    c. 18

    d. -25

    24. Find the dot product of B1 = 3i – 3j + 7k; B2 = 9i + 3j – 4k  

    a. 8

    b. -9

    c. -10

    d. -8

    25. What is the determinant of A:  

    2

    a. 70

    b. 75

    c. 80

    d. 85

    26. Find h(x) if h[f(x)] = (2x2-1)5 and f(x) = 2x2-1 

    a. x5

    b. 2x2

    c. –x5

    d. -2x5

    27. Find the inverse of the function g(x) = 3x + 2

    a. (y-1)/2

    b. (y+3)/2

    c. (y+2)/3

    d. (y-2)3

    28. Let f(x)=(x-3)1/3 and h(x) = x3, find h[f(x)]

    a. x-2

    b. x-3

    c. x+3

    d. x-1

    29. Let f(x) = (x-3)1/3 and h(x) = x3 , find f[h(x)]

    a. (x2-2)1/3

    b. (x2+2)1/3

    c. (x2-3)1/3

    d. (x2+3)1/3

    30. For f(x) = (x-2)0.5, find f(3x)

    a. (5x-2)0.5

    b. (2x-1)0.5

    c. (3x+2)0.5

    d. (3x-2)0.5

    31. For f(x) = 3x – 2, find f(2)

    a. 4

    b. 3

    c. 5

    d. 2

    32. Find x: x = (7x – 10)0.5

    a. 5,2

    b. 5,-3

    c. 2,-5

    d. 2,4

    33. Solve the equation simultaneously: 2x2-3y2-14=0 and y=x-2 

    a. (9,7), (3,1)

    b. (10,8), (-3,2)

    c. (9,-7),(3,2)

    d. (9,7),(-3,-1)

    34. Solve algebraically x and y: 3x2+8y2=35 and 12y2-4x2=42 

    a. (1,2)

    b. (±1,±2)

    c. (-1,-2)

    d. (-1,2)

    35. Determine the value of m so that the equation 9x2 + mx + 1 = 0 , will just one real solution.

    a. ±6

    b. ±7

    c. ±5

    d. ±8

    36. Solve for y: y2-3y-10=0 

    a. -2,-5

    b. -2,5

    c. 5,0

    d. -2,8

    37. Evaluate (3 + i)5

    a. 12+315i

    b. -12+300i

    c. -12+316i

    d. -10-310i

    38. Express 4 + 8i in polar form.

    a. 9.32 , 65.22o

    b. 10.32 , 60.22o

    c. 8.39 , 60.28o

    d. 8.94 , 63.43o

    39. What is the product of the complex numbers 5 + 8i and 7 – 3i?

    a. 59+41i

    b. 62+40i

    c. 50+35i

    d. 59+48i

    40. What is the exponential form of the complex number 1 + 5i?

    a. 5.1e78.7i

    b. 6.3e75.2i

    c. 4.3e70.2i

    d. 3.2e68.7i

    41. In the expansion of (x – 3y)7 find the y3-term, that is, the term involving y3.

    a. 949x4y3

    b. -940x4y3

    c. -945x4y3

    d. 900x4y3

    42. Find the 3rd term of the expansion of (3x – y)5.

    a. -270x3y2

    b. 270x3y2

    c. 270x2y2

    d. -270x2y2

    43. Find the sum of coefficients of (3x – 4y)12

    a. 4

    b. 3

    c. 2

    d. 1

    44. Find the constant term in the expansion of [3x3 – (1/x2)]5

    a. -90

    b. -92

    c. -91

    d. -89

    45. Resolve the following into partial fractions: (3x2 – 8x + 9)/(x-2)3

    a. [3/(x-2)] + [4/(x-2)2] + [5/(x-2)3]

    b. [3/(x+1)] – [2/(x+1)2] + [4/(x+1)3]

    c. [2/(x+1)] + [4/(x+1)2] + [3/(x+1)3]

    d. [4/(x+1)] – [2/(x+1)2] – [3/(x+1)3]

    Answer Key (Handwritten)

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