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6.E: Review Exercises and Sample Exam - Mathematics


Review Exercises

Exercise (PageIndex{1}) Introduction to Factoring

Determine the missing factor.

  1. (12x^{3}−24x^{2}+4x=4x( ? ))
  2. (10y^{4}−35y^{3}−5y^{2}=5y^{2}( ? ))
  3. (−18a^{5}+9a^{4}−27a^{3}=−9a^{3}( ? ))
  4. (−21x^{2}y+7xy^{2}−49xy=−7xy( ? ))
Answer

1. ((3x^{2}−6x+1))

3. ((2a^{2}−a+3))

Exercise (PageIndex{2}) Introduction to Factoring

Factor out the GCF.

  1. (22x^{2}+11x)
  2. (15y^{4}−5y^{3})
  3. (18a^{3}−12a^{2}+30a)
  4. (12a^{5}+20a^{3}−4a)
  5. (9x^{3}y^{2}−18x^{2}y^{2}+27xy^{2})
  6. (16a^{5}b^{5}c−8a^{3}b^{6}+24a^{3}b^{2}c)
Answer

1. (11x(2x+1))

3. (6a(3a^{2}−2a+5))

5. (9xy2(x^{2}−2x+3))

Exercise (PageIndex{3}) Introduction to Factoring

Factor by grouping.

  1. (x^{2}+2x−5x−10 )
  2. (2x^{2}−2x−3x+3 )
  3. (x^{3}+5x^{2}−3x−15 )
  4. (x^{3}−6x^{2}+x−6 )
  5. (x^{3}−x^{2}y−2x+2y )
  6. (a^{2}b^{2}−2a^{3}+6ab−3b^{3})
Answer

1. ((x+2)(x−5))

3. ((x+5)(x^{2}−3))

5. ((x−y)(x^{2}−2))

Exercise (PageIndex{4}) Factoring Trinomials of the Form (x^{2}+bx+c)

Are the following factored correctly? Check by multiplying.

  1. (x^{2}+5x+6=(x+6)(x−1) )
  2. (x^{2}+3x−10=(x+5)(x−2) )
  3. (x^{2}+6x+9=(x+3)^{2} )
  4. (x^{2}−6x−9=(x−3)(x+3))
Answer

1. No

3. Yes

Exercise (PageIndex{5}) Factoring Trinomials of the Form (x^{2}+bx+c)

Factor.

  1. (x^{2}−13x−14 )
  2. (x^{2}+13x+12 )
  3. (y^{2}+10y+25 )
  4. (y^{2}−20y+100 )
  5. (a^{2}−8a−48 )
  6. (b^{2}−18b+45 )
  7. (x^{2}+2x+24 )
  8. (x^{2}−10x−16 )
  9. (a^{2}+ab−2b^{2} )
  10. (a^{2}b^{2}+5ab−50)
Answer

1. ((x−14)(x+1))

3. ((y+5)^{2})

5. ((a−12)(a+4))

7. Prime

9. ((a−b)(a+2b))

Exercise (PageIndex{6}) Factoring Trinomials of the Form (ax^{2}+bx+c)

Factor.

  1. (5x^{2}−27x−18 )
  2. (3x^{2}−14x+8 )
  3. (4x^{2}−28x+49 )
  4. (9x^{2}+48x+64 )
  5. (6x^{2}−29x−9 )
  6. (8x^{2}+6x+9 )
  7. (60x^{2}−65x+15 )
  8. (16x^{2}−40x+16 )
  9. (6x^{3}−10x^{2}y+4xy^{2})
  10. (10x^{3}y−82x^{2}y^{2}+16xy^{3})
  11. (−y^{2}+9y+36 )
  12. (−a^{2}−7a+98 )
  13. (16+142x−18x^{2} )
  14. (45−132x−60x^{2})
Answer

1. ((5x+3)(x−6) )

3. ((2x−7)^{2})

5. Prime

7. (5(3x−1)(4x−3) )

9. (2x(3x−2y)(x−y) )

11. (−1(y−12)(y+3) )

13. (−2(9x+1)(x−8))

Exercise (PageIndex{7}) Factoring Special Binomials

Factor completely.

  1. (x^{2}−81)
  2. (25x^{2}−36)
  3. (4x^{2}−49)
  4. (81x^{2}−1)
  5. (x^{2}−64y^{2})
  6. (100x^{2}y^{2}−1)
  7. (16x^{4}−y^{4})
  8. (x^{4}−81y^{4})
  9. (8x^{3}−125)
  10. (27+y^{3})
  11. (54x^{4}y−2xy^{4})
  12. (3x^{4}y^{2}+24xy^{5})
  13. (64x^{6}−y^{6})
  14. (x^{6}+1)
Answer

1. ((x+9)(x−9))

3. ((2x+7)(2x−7))

5. ((x+8y)(x−8y))

7. ((4x^{2}+y^{2})(2x+y)(2x−y))

9. ((2x−5)(4x^{2}+10x+25))

11. (2xy(3x−y)(9x^{2}+3xy+y^{2}))

13. ((2x+y)(4x^{2}−2xy+y^{2})(2x−y)(4x^{2}+2xy+y^{2}))

Exercise (PageIndex{8}) General Guidelines for Factoring Polynomials

Factor completely.

  1. (8x^{3}−4x^{2}+20x)
  2. (50a^{4}b^{4}c+5a^{3}b^{5}c^{2})
  3. (x^{3}−12x^{2}−x+12)
  4. (a^{3}−2a^{2}−3ab+6b)
  5. (−y^{2}−15y+16)
  6. (x^{2}−18x+72)
  7. (144x^{2}−25)
  8. (3x^{4}−48)
  9. (20x^{2}−41x−9)
  10. (24x^{2}+14x−20)
  11. (a^{4}b−343ab^{4})
  12. (32x^{7}y^{2}+4xy^{8})
Answer

1. (4x(2x^{2}−x+5))

3. ((x−12)(x+1)(x−1))

5. (−1(y+16)(y−1))

7. ((12x+5)(12x−5))

9. ((4x−9)(5x+1))

11. (ab(a−7b)(a^{2}+7ab+49b^{2}))

Exercise (PageIndex{9}) Solving Equations by Factoring

Solve.

  1. ((x−9)(x+10)=0 )
  2. (−3x(x+8)=0 )
  3. (6(x+1)(x−1)=0 )
  4. ((x−12)(x+4)(2x−1)=0 )
  5. (x^{2}+5x−50=0 )
  6. (3x^{2}−13x+4=0 )
  7. (3x^{2}−12=0 )
  8. (16x^{2}−9=0 )
  9. ((x−2)(x+6)=20 )
  10. (2(x−2)(x+3)=7x−9 )
  11. (52x^{2}−203x=0 )
  12. (23x^{2}−512x+124=0)
Answer

1. (9, −10)

3. (−1, 1)

5. (−10, 5)

7. (±2)

9. (−8, 4)

11. (0, frac{8}{3})

Exercise (PageIndex{10}) Solving Equations by Factoring

Find a quadratic equation with integer coefficients, given the following solutions.

  1. (−7, 6)
  2. (0, −10)
  3. (−frac{1}{9}, frac{1}{2})
  4. (± frac{3}{2})
Answer

1. (x^{2}+x−42=0)

3. (18x^{2}−7x−1=0)

Exercise (PageIndex{11}) Applications Involving Quadratic Equations

Set up an algebraic equation and then solve the following.

  1. An integer is (4) less than twice another. If the product of the two integers is (96), then find the integers.
  2. The sum of the squares of two consecutive positive even integers is (52). Find the integers.
  3. A (20)-foot ladder leaning against a wall reaches a height that is (4) feet more than the distance from the wall to the base of the ladder. How high does the ladder reach?
  4. The height of an object dropped from the top of a (196)-foot building is given by (h(t)=−16t^{2}+196), where (t) represents the number of seconds after the object has been released. How long will it take the object to hit the ground?
  5. The length of a rectangle is (1) centimeter less than three times the width. If the area is (70) square centimeters, then find the dimensions of the rectangle.
  6. The base of a triangle is (4) centimeters more than twice the height. If the area of the triangle is (80) square centimeters, then find the measure of the base.
Answer

1. {(8, 12)} or {(−6, −16)}

3. (16) feet

5. Length: (14) centimeters; width: (5) centimeters

Sample Exam

Exercise (PageIndex{12})

  1. Determine the GCF of the terms (25a^{2}b^{2}c, 50ab^{4}), and (35a^{3}b^{3}c^{2}).
  2. Determine the missing factor: (24x^{2}y^{3}−16x^{3}y^{2}+8x^{2}y=8x^{2}y( ? )).
Answer

1. (5ab^{2})

Exercise (PageIndex{13})

Factor.

  1. (12x^{5}−15x^{4}+3x^{2})
  2. (x^{3}−4x^{2}−2x+8)
  3. (x^{2}−7x+12)
  4. (9x^{2}−12x+4)
  5. (x^{2}−81)
  6. (x^{3}+27y^{3})
Answer

1. (3x^{2}(4x^{3}−5x^{2}+1))

3. ((x−4)(x−3) )

5. ((x+9)(x−9))

Exercise (PageIndex{14})

Factor completely.

  1. (x^{3}+2x^{2}−4x−8)
  2. (x^{4}−1)
  3. (−6x^{3}+20x^{2}−6x)
  4. (x^{6}−1)
Answer

1. ((x+2)^{2}(x−2))

3. (−2x(3x−1)(x−3))

Exercise (PageIndex{15})

Solve.

  1. ((2x+1)(x−7)=0 )
  2. (3x(4x−3)(x+1)=0 )
  3. (x^{2}−64=0 )
  4. (x^{2}+4x−12=0 )
  5. (23x^{2}+89x−16=0 )
  6. ((x−5)(x−3)=−1 )
  7. (3x(x+3)=14x+2 )
  8. ((3x+1)(3x+2)=9x+3)
Answer

1. (−frac{1}{2}, 7 )

3. (±8 )

5. (−frac{3}{2}, frac{1}{6})

7. (−frac{1}{3}, 2)

Exercise (PageIndex{16})

For each problem, set up an algebraic equation and then solve.

  1. An integer is (4) less than twice another. If the product of the two integers is (70), then find the integers.
  2. The sum of the squares of two consecutive positive odd integers is (130). Find the integers.
  3. The length of a rectangle is (4) feet more than twice its width. If the area is (160) square feet, then find the dimensions of the rectangle.
  4. The height of a triangle is (6) centimeters less than four times the length of its base. If the area measures (27) square centimeters, then what is the height of the triangle?
  5. The height of a projectile launched upward at a speed of (64) feet/second from a height of (36) feet is given by the function (h(t)=−16t^{2}+64t+36). How long will it take the projectile to hit the ground?
Answer

1. {(7, 10)} or {(−14, −5)}

3. Width: (8) feet; length: (20) feet

5. (4frac{1}{2}) sec


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      math-2022/5697/image_dhDgxYKCK2w2wB.png

      Getting Started
      The College Placement Math Plan 9
      Making a Study Schedule 10

      Basic Math
      Fraction Tips, Tricks and Shortcuts 16
      Converting Fractions to Decimals 18
      Converting Fractions to Percent 20
      Decimal Tips, Tricks and Shortcuts 22
      Converting Decimals to Fractions 22
      Converting Decimals to Percent 22
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      Types of Word Problems 55
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      Answer Key 77

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      Pythagorean Geometry 92
      Quadrilaterals 95
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      Factoring Polynomials 144
      Quadratic Equations 146
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