6.E: Review Exercises and Sample Exam
- Page ID
- 23746
Review Exercises
Exercise \(\PageIndex{1}\) Introduction to Factoring
Determine the missing factor.
- \(12x^{3}−24x^{2}+4x=4x( ? )\)
- \(10y^{4}−35y^{3}−5y^{2}=5y^{2}( ? )\)
- \(−18a^{5}+9a^{4}−27a^{3}=−9a^{3}( ? )\)
- \(−21x^{2}y+7xy^{2}−49xy=−7xy( ? )\)
- Answer
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1. \((3x^{2}−6x+1)\)
3. \((2a^{2}−a+3)\)
Exercise \(\PageIndex{2}\) Introduction to Factoring
Factor out the GCF.
- \(22x^{2}+11x\)
- \(15y^{4}−5y^{3}\)
- \(18a^{3}−12a^{2}+30a\)
- \(12a^{5}+20a^{3}−4a\)
- \(9x^{3}y^{2}−18x^{2}y^{2}+27xy^{2}\)
- \(16a^{5}b^{5}c−8a^{3}b^{6}+24a^{3}b^{2}c\)
- Answer
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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.
- \(x^{2}+2x−5x−10 \)
- \(2x^{2}−2x−3x+3 \)
- \(x^{3}+5x^{2}−3x−15 \)
- \(x^{3}−6x^{2}+x−6 \)
- \(x^{3}−x^{2}y−2x+2y \)
- \(a^{2}b^{2}−2a^{3}+6ab−3b^{3}\)
- Answer
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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.
- \(x^{2}+5x+6=(x+6)(x−1) \)
- \(x^{2}+3x−10=(x+5)(x−2) \)
- \(x^{2}+6x+9=(x+3)^{2} \)
- \(x^{2}−6x−9=(x−3)(x+3)\)
- Answer
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1. No
3. Yes
Exercise \(\PageIndex{5}\) Factoring Trinomials of the Form \(x^{2}+bx+c\)
Factor.
- \(x^{2}−13x−14 \)
- \(x^{2}+13x+12 \)
- \(y^{2}+10y+25 \)
- \(y^{2}−20y+100 \)
- \(a^{2}−8a−48 \)
- \(b^{2}−18b+45 \)
- \(x^{2}+2x+24 \)
- \(x^{2}−10x−16 \)
- \(a^{2}+ab−2b^{2} \)
- \(a^{2}b^{2}+5ab−50\)
- Answer
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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.
- \(5x^{2}−27x−18 \)
- \(3x^{2}−14x+8 \)
- \(4x^{2}−28x+49 \)
- \(9x^{2}+48x+64 \)
- \(6x^{2}−29x−9 \)
- \(8x^{2}+6x+9 \)
- \(60x^{2}−65x+15 \)
- \(16x^{2}−40x+16 \)
- \(6x^{3}−10x^{2}y+4xy^{2}\)
- \(10x^{3}y−82x^{2}y^{2}+16xy^{3}\)
- \(−y^{2}+9y+36 \)
- \(−a^{2}−7a+98 \)
- \(16+142x−18x^{2} \)
- \(45−132x−60x^{2}\)
- Answer
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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.
- \(x^{2}−81\)
- \(25x^{2}−36\)
- \(4x^{2}−49\)
- \(81x^{2}−1\)
- \(x^{2}−64y^{2}\)
- \(100x^{2}y^{2}−1\)
- \(16x^{4}−y^{4}\)
- \(x^{4}−81y^{4}\)
- \(8x^{3}−125\)
- \(27+y^{3}\)
- \(54x^{4}y−2xy^{4}\)
- \(3x^{4}y^{2}+24xy^{5}\)
- \(64x^{6}−y^{6}\)
- \(x^{6}+1\)
- Answer
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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.
- \(8x^{3}−4x^{2}+20x\)
- \(50a^{4}b^{4}c+5a^{3}b^{5}c^{2}\)
- \(x^{3}−12x^{2}−x+12\)
- \(a^{3}−2a^{2}−3ab+6b\)
- \(−y^{2}−15y+16\)
- \(x^{2}−18x+72\)
- \(144x^{2}−25\)
- \(3x^{4}−48\)
- \(20x^{2}−41x−9\)
- \(24x^{2}+14x−20\)
- \(a^{4}b−343ab^{4}\)
- \(32x^{7}y^{2}+4xy^{8}\)
- Answer
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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.
- \((x−9)(x+10)=0 \)
- \(−3x(x+8)=0 \)
- \(6(x+1)(x−1)=0 \)
- \((x−12)(x+4)(2x−1)=0 \)
- \(x^{2}+5x−50=0 \)
- \(3x^{2}−13x+4=0 \)
- \(3x^{2}−12=0 \)
- \(16x^{2}−9=0 \)
- \((x−2)(x+6)=20 \)
- \(2(x−2)(x+3)=7x−9 \)
- \(52x^{2}−203x=0 \)
- \(23x^{2}−512x+124=0\)
- Answer
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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.
- \(−7, 6\)
- \(0, −10\)
- \(−\frac{1}{9}, \frac{1}{2}\)
- \(± \frac{3}{2}\)
- 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.
- An integer is \(4\) less than twice another. If the product of the two integers is \(96\), then find the integers.
- The sum of the squares of two consecutive positive even integers is \(52\). Find the integers.
- 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?
- 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?
- 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.
- 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 \(\PageIndex{12}\)
- Determine the GCF of the terms \(25a^{2}b^{2}c, 50ab^{4}\), and \(35a^{3}b^{3}c^{2}\).
- 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.
- \(12x^{5}−15x^{4}+3x^{2}\)
- \(x^{3}−4x^{2}−2x+8\)
- \(x^{2}−7x+12\)
- \(9x^{2}−12x+4\)
- \(x^{2}−81\)
- \(x^{3}+27y^{3}\)
- Answer
-
1. \(3x^{2}(4x^{3}−5x^{2}+1)\)
3. \((x−4)(x−3) \)
5. \((x+9)(x−9)\)
Exercise \(\PageIndex{14}\)
Factor completely.
- \(x^{3}+2x^{2}−4x−8\)
- \(x^{4}−1\)
- \(−6x^{3}+20x^{2}−6x\)
- \(x^{6}−1\)
- Answer
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1. \((x+2)^{2}(x−2)\)
3. \(−2x(3x−1)(x−3)\)
Exercise \(\PageIndex{15}\)
Solve.
- \((2x+1)(x−7)=0 \)
- \(3x(4x−3)(x+1)=0 \)
- \(x^{2}−64=0 \)
- \(x^{2}+4x−12=0 \)
- \(23x^{2}+89x−16=0 \)
- \((x−5)(x−3)=−1 \)
- \(3x(x+3)=14x+2 \)
- \((3x+1)(3x+2)=9x+3\)
- Answer
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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.
- An integer is \(4\) less than twice another. If the product of the two integers is \(70\), then find the integers.
- The sum of the squares of two consecutive positive odd integers is \(130\). Find the integers.
- 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.
- 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?
- 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
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1. {\(7, 10\)} or {\(−14, −5\)}
3. Width: \(8\) feet; length: \(20\) feet
5. \(4\frac{1}{2}\) sec