Practice Makes Perfect
Recognize a Preliminary Strategy to Factor Polynomials Completely
In the following exercises, identify the best method to use to factor each polynomial.
Exercise 1
- \(10q^2+50\)
- \(a^2−5a−14\)
- \(uv+2u+3v+6\)
- Answer
-
- factor the GCF, binomial
- Undo FOIL
- factor by grouping
Exercise 2
- \(n^2+10n+24\)
- \(8u^2+16\)
- \(pq+5p+2q+10\)
Exercise 3
- \(x^2+4x−21\)
- \(ab+10b+4a+40\)
- \(6c^2+24\)
- Answer
-
- undo FOIL
- factor by grouping
- factor the GCF, binomial
Exercise 4
- \(20x^2+100\)
- \(uv+6u+4v+24\)
- \(y^2−8y+15\)
Factor Trinomials of the form \(ax^2+bx+c\) with a GCF
In the following exercises, factor completely.
Exercise 5
\(5x^2+35x+30\)
- Answer
-
\(5(x+1)(x+6)\)
Exercise 6
\(12s^2+24s+12\)
Exercise 7
\(2z^2−2z−24\)
- Answer
-
\(2(z−4)(z+3)\)
Exercise 8
\(3u^2−12u−36\)
Exercise 9
\(7v^2−63v+56\)
- Answer
-
\(7(v−1)(v−8)\)
Exercise 10
\(5w^2−30w+45\)
Exercise 11
\(p^3−8p^2−20p\)
- Answer
-
\(p(p−10)(p+2)\)
Exercise 12
\(q^3−5q^2−24q\)
Exercise 13
\(3m^3−21m^2+30m\)
- Answer
-
\(3m(m−5)(m−2)\)
Exercise 14
\(11n^3−55n^2+44n\)
Exercise 15
\(5x^4+10x^3−75x^2\)
- Answer
-
\(5x^{2}(x−3)(x+5)\)
Exercise 16
\(6y^4+12y^3−48y^2\)
Factor Trinomials Using Trial and Error
In the following exercises, factor.
Exercise 17
\(2t^2+7t+5\)
- Answer
-
\((2t+5)(t+1)\)
Exercise 18
\(5y^2+16y+11\)
Exercise 19
\(11x^2+34x+3\)
- Answer
-
\((11x+1)(x+3)\)
Exercise 20
\(7b^2+50b+7\)
Exercise 21
\(4w^2−5w+1\)
- Answer
-
\((4w−1)(w−1)\)
Exercise 22
\(5x^2−17x+6\)
Exercise 23
\(6p^2−19p+10\)
- Answer
-
\((3p−2)(2p−5)\)
Exercise 24
\(21m^2−29m+10\)
Exercise 25
\(4q^2−7q−2\)
- Answer
-
\((4q+1)(q−2)\)
Exercise 26
\(10y^2−53y−11\)
Exercise 27
\(4p^2+17p−15\)
- Answer
-
\((4p−3)(p+5)\)
Exercise 28
\(6u^2+5u−14\)
Exercise 29
\(16x^2−32x+16\)
- Answer
-
\(16(x−1)(x−1)\)
Exercise 30
\(81a^2+153a−18\)
Exercise 31
\(30q^3+140q^2+80q\)
- Answer
-
\(10q(3q+2)(q+4)\)
Exercise 32
\(5y^3+30y^2−35y\)
Factor Trinomials using the ‘ac’ Method
In the following exercises, factor.
Exercise 33
\(5n^2+21n+4\)
- Answer
-
\((5n+1)(n+4)\)
Exercise 34
\(8w^2+25w+3\)
Exercise 35
\(9z^2+15z+4\)
- Answer
-
\((3z+1)(3z+4)\)
Exercise 36
\(3m^2+26m+48\)
Exercise 37
\(4k^2−16k+15\)
- Answer
-
\((2k−3)(2k−5)\)
Exercise 38
\(4q^2−9q+5\)
Exercise 39
\(5s^2−9s+4\)
- Answer
-
\((5s−4)(s−1)\)
Exercise 40
\(4r^2−20r+25\)
Exercise 41
\(6y^2+y−15\)
- Answer
-
\((3y+5)(2y−3)\)
Exercise 42
\(6p^2+p−22\)
Exercise 43
\(2n^2−27n−45\)
- Answer
-
\((2n+3)(n−15)\)
Exercise 44
\(12z^2−41z−11\)
Exercise 45
\(3x^2+5x+4\)
- Answer
-
prime
Exercise 46
\(4y^2+15y+6\)
Exercise 47
\(60y^2+290y−50\)
- Answer
-
\(10(6y−1)(y+5)\)
Exercise 48
\(6u^2−46u−16\)
Exercise 49
\(48z^3−102z^2−45z\)
- Answer
-
\(3z(8z+3)(2z−5)\)
Exercise 50
\(90n^3+42n^2−216n\)
Exercise 51
\(16s^2+40s+24\)
- Answer
-
\(8(2s+3)(s+1)\)
Exercise 52
\(24p^2+160p+96\)
Exercise 53
\(48y^2+12y−36\)
- Answer
-
\(12(4y−3)(y+1)\)
Exercise 54
\(30x^2+105x−60\)
Mixed Practice
In the following exercises, factor.
Exercise 55
\(12y^2−29y+14\)
- Answer
-
\((4y−7)(3y−2)\)
Exercise 56
\(12x^2+36y−24z\)
Exercise 57
\(a^2−a−20\)
- Answer
-
\((a−5)(a+4)\)
Exercise 59
\(6n^2+5n−4\)
- Answer
-
\((2n−1)(3n+4)\)
Exercise 60
\(12y^2−37y+21\)
Exercise 61
\(2p^2+4p+3\)
- Answer
-
prime
Exercise 62
\(3q^2+6q+2\)
Exercise 63
\(13z^2+39z−26\)
- Answer
-
\(13(z^2+3z−2)\)
Exercise 64
\(5r^2+25r+30\)
Exercise 65
\(x^2+3x−28\)
- Answer
-
\((x+7)(x−4)\)
Exercise 66
\(6u^2+7u−5\)
Exercise 67
\(3p^2+21p\)
- Answer
-
\(3p(p+7)\)
Exercise 69
\(6r^2+30r+36\)
- Answer
-
\(6(r+2)(r+3)\)
Exercise 70
\(18m^2+15m+3\)
Exercise 71
\(24n^2+20n+4\)
- Answer
-
\(4(2n+1)(3n+1)\)
Exercise 72
\(4a^2+5a+2\)
Exercise 73
\(x^2+2x−24\)
- Answer
-
\((x+6)(x−4)\)
Exercise 74
\(2b^2−7b+4\)
Writing Exercises
Exercise 77
List, in order, all the steps you take when using the “\(ac\)” method to factor a trinomial of the form \(ax^2+bx+c\).
- Answer
-
Answers may vary.
Exercise 78
How is the “\(ac\)” method similar to the “undo FOIL” method? How is it different?
Exercise 79
What are the questions, in order, that you ask yourself as you start to factor a polynomial? What do you need to do as a result of the answer to each question?
- Answer
-
Answers may vary.
Exercise 80
On your paper draw the chart that summarizes the factoring strategy. Try to do it without looking at the book. When you are done, look back at the book to finish it or verify it.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?