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Mathematics LibreTexts

6.E: Review Exercises and Sample Exam

  • Page ID
    23746
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    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. \(4\frac{1}{2}\) sec