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Mathematics LibreTexts

0.05e: Exercises - Factoring

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A: Factor out the GCF

Exercise 0.5.A 

 Factor out the GCF.

  1. 12x416x3+4x2
  2. 15x510x45x3
  3. 20y8+28y6+40y3
  4. 18y724y530y3
  5. 2a4b36a3b2+8a2b
  6. 28a3b321a2b414ab5
  7. 2x3y54x4y4+x2y3
  1. 3x5y2x4y2+x3y3
  2. 5x2(2x+3)3(2x+3)
  3. y2(y1)+9(y1)
  4. 9x2(3x1)+(3x1)
  5. 7y2(5y+2)(5y+2)
  6. x5nx3n+xn
  7. y6ny3ny2n
  1. 2x412x32x2
  2. 18a3b3a2b2+3ab3
  3. x4y33x3y+x2y
  4. x3nx2nxn
  5. 35y+84
  6. 6y2+12y6
  1. 18x315x
  2. 15m4+6m2n
  3. 4x312x2+16x
  4. 3x+24
  5. 3x3+27x212x
  6. 3x(x1)+5(x1)
Answers to odd exercises:
1. 4x2(3x24x+1)
3. 4y3(5y5+7y3+10)
5. 2a2b(a2b23ab+4)
7. x2y3(2xy24x2y+1)
9. (2x+3)(5x23)
11. (3x1)(9x2+1)
13. xn(x4nx2n+1)
15. 2x2(x26x1)
17. x2y(x2y23x+1) 
19. 7(5y+12)
21. 3x(6x25)
23. 4x(x23x+4)
25. 3x(x29x+4)

B: Factor 4 term Polynomials.

Exercise 0.5.B 

 Factor by grouping.

  1. 2x3+3x2+2x+3
  2. 5x3+25x2+x+5
  3. 6x33x2+4x2
  4. 3x32x215x+10
  5. x3x23x+3
  6. 6x315x22x+5
  7. 2x3+7x210x35
  8. 3x3x2+24x8
  9. 14y4+10y37y5
  10. 5y4+2y3+20y+8
  11. x4n+x3n+2xn+2
  12. x5n+x3n+3x2n+3
  13. x3x2y+xy2y3
  14. x3+x2y2xy22y3
  15. 3x3y2+9x2y3x3y
  16. 2x3y3x2y3+2xy
  17. a2b4ab23a+12b
  1. a2b+3ab2+5a+15b
  2. a4+a2b3+a2b+b4
  3. a3b+2a2+3ab4+6b3
  4. 3ax+10by5ay6bx
  5. a2x5b2y5a2y+b2x
  6. x4y2x3y3+x2y4xy5
  7. 2x5y2+4x4y2+18x3y+36x2y
  8. a5b2+a4b4+a3b3+a2b5
  9. 3a6b+3a5b2+9a4b2+9a3b3
  10. 2x3x2+2x1
  11. 3x3x26x+2
  12. x35x2y+xy25y3
  13. a2ba+ab3b2
  14. 2x44xy3+2x2y24x3y
  15. x4y2xy5+x3y4x2y3
  16. axay+bxby
  17. x2yxy2+2x2y
  1. x2+7x3x21
  2. 4x216x+3x12
  3. m3+m2+m+1
  4. 5x5yy+x  
  5. 9x2+18x+9121x2
  6. 9x2+30x+2581x2
  7. 16x2+24x+9144x2
  8. 16x2+32x+16121x2
  9. 25x2+70x+49121x2
  10. 25x2+80x+64100x2
  11. 9x2+12x+4144
  12. 9x2+24x+16144
  13. 16x2+24x+936
  14. 16x2+48x+3681
  15. 25x2+30x+981
  16. 25x2+20x+4100
Answers to odd exercises:
31. (2x+3)(x2+1)
33. (2x1)(3x2+2)
35. (x1)(x23)
37. (2x+7)(x25)
39. (7y+5)(2y31) 
41. (xn+1)(x3n+2)
43. (xy)(x2+y2)
45. (x+3y)(3x2y21)
47. (a4b)(ab3)
49. (a2+b)(a2+b3)
51. (a2b)(3x5y)
53. xy2(xy)(x2+y2)
55. a2b2(a2+b)(a+b2)
57. (x2+1)(2x1)
59. (x2+y2)(x5y)
61. 2x(x2y)(x2+y2)
63. (a+b)(xy)
65. (x3)(x+7)
67. (m2+1)(m+1)
69. (8x+3)(14x+3)
71. (8x+3)(16x+3)
73. (6x+7)(16x+7)
75. (3x10)(3x+14)
77. (4x3)(4x+9)
79. (5x6)(5x+12)

C: Factor Binomials.

Exercise 0.5.C 

 I. Factor two term polynomials.

  1. x264
  2. x2100
  3. 94y2
  4. 25y2
  5. x281y2
  6. x249y2
  7. a2b24
  8. 19a2b2
  9. a2b2c2
  10. 4a2b2c2
  1. x464
  2. 36y4
  3. (2x+5)2x2
  4. (3x5)2x2
  5. y2(y3)2
  6. y2(2y+1)2
  7. (2x+5)2(x3)2
  8. (3x1)2(2x3)2
  9. x416
  10. 81x41
  1. x4y41
  2. x4y4
  3. x8y8
  4. y81
  5. x2ny2n
  6. x2ny2n4
  7. x4ny4n
  8. x4ny4n16
  9. x327
  10. 8x3125
  1. 8y3+27
  2. 64x3+343
  3. x3y3
  4. x3+y3
  5. 8a3b3+1
  6. 27a38b3
  7. x3y3125
  8. 216x3+y3
  9. x3+(x+3)3
  10. y3(2y1)3
  1. (2x+1)3x3
  2. (3y5)3y3
  3. x3ny3n
  4. x3n+y3n
  5. a6+64
  6. 64a61
  7. x6y6
  8. x6+y6
  9. x6ny6n
  10. x6n+y6n
Answers to odd exercises:
81. (x+8)(x8)
83. (3+2y)(32y)
85. (x+9y)(x9y)
87. (ab+2)(ab2)
89. (ab+c)(abc)
91. (x2+8)(x28)
93. (3x+5)(x+5)
95. 3(2y3)
97. (3x+2)(x+8)
99. (x2+4)(x+2)(x2)
101. (x2y2+1)(xy+1)(xy1)
103. (x4+y4)(x2+y2)(x+y)(xy)
105. (xn+yn)(xnyn)
107. (x2n+y2n)(xn+yn)(xnyn)
109. (x3)(x2+3x+9)
111. (2y+3)(4y26y+9)
113. (xy)(x2+xy+y2)
115. (2ab+1)(4a2b22ab+1)
117. (xy5)(x2y2+5xy+25)
119. (2x+3)(x2+3x+9)
121. (x+1)(7x2+5x+1)
123. (xnyn)(x2n+xnyn+y2n)
125. (a2+4)(a44a2+16)
127. (x+y)(x2xy+y2)(xy)(x2+xy+y2)
129. (xn+yn)(x2nxnyn+y2n)(xnyn)(x2n+xnyn+y2n)

 II. Factor two term polynomials.

  1. 64x21
  2. 9100y2
  3. x236y2
  4. 4(2x1)2
  5. a3b3+125
  6. 64x3y3
  1. 81x4y4
  2. x81
  3. x664y6
  4. 1a6b6
  5. 81x29x
  6. 6416x
  1. 169m2n2
  2. 25p21
  3. 9121y2
  4. 20x2125
  5. 169n3n
  6. 6p2q254p2
  1. 24p2+54
  2. 49x281y2
  3. 16z41
  4. 48m4n2243n2
  5. a2+6a+99b2
  6. x216x+64y2
  1. a3125
  2. b3216
  3. 2m3+54
  4. 81m3+3
Answers to odd exercises:
131. (8x+1)(8x1)
133. (x+6y)(x6y)
135. (ab+5)(a2b25ab+25)
137. (9x2+y2)(3x+y)(3xy)
139. (x+2y)(x22xy+4y2)(x2y)(x2+2xy+4y2)
141. 9x(9x1)
143. (13m+n)(13mn)
145. (3+11y)(311y)
147. n(13n+1)(13n1)
149. 6(4p2+9)
151. (2z1)(2z+1)(4z2+1)
153. (a+33b)(a+3+3b)
155. (a5)(a2+5a+25)
157. 2(m+3)(m23m+9)
 

D: Factor Trinomials, a=1.

Exercise 0.5.D 

 Factor trinomials, a=1.

  1. x2+5x6
  2. x2+5x+6
  3. x2+4x12
  4. x2+3x18
  5. x214x+48
  6. x215x+54
  7. x2+11x30
  8. x22x+24
  9. x218x+81
  10. x222x+121
  1. x2xy20y2
  2. x2+10xy+9y2
  3. x2y2+5xy50
  4. x2y216xy+48
  5. a26ab72b2
  6. a221ab+80b2
  7. u2+14uv32v2
  8. m2+7mn98n2
  9. (x+y)22(x+y)8
  10. (xy)22(xy)15
  1. x47x28
  2. x4+13x2+30
  3. x48x248
  4. x4+25x2+24
  5. y420y2+100
  6. y4+14y2+49
  7. x4+3x2y2+2y4
  8. x48x2y2+15y4
  9. a4b44a2b2+4
  10. a4+6a2b2+9b4
  1. x618x340
  2. x6+18x3+45
  3. x6x3y36y6
  4. x6+x3y320y6
  5. x6y6+2x3y315
  6. x6y6+16x3y3+48
  7. x2n+12xn+32
  8. x2n+41xn+40
  9. x2n+2axn+a2
  10. x2n2axn+a2
Answers to odd exercises:
161. (x1)(x+6)
163. (x2)(x+6)
165. (x6)(x8)
167. Prime
169. (x9)2
171. (x5y)(x+4y)
173. (xy5)(xy+10)
175. (a+6b)(a12b)
177. (u2v)(u+16v)
179. (x+y4)(x+y+2)
181. (x28)(x2+1)
183. (x2+4)(x212)
185. (y210)2
187. (x2+y2)(x2+2y2)
189. (a2b22)2 
191. (x320)(x3+2)
193. (x3+2y3)(x33y3)
195. (x3y33)(x3y3+5)
197. (xn+4)(xn+8)
199. (xn+a)2

E: Factor trinomials, a1

Exercise 0.5.E 

 Factor trinomials. (Leading coefficient is not one).

  1. 3x2+20x7
  2. 2x29x5
  3. 6a2+13a+6
  4. 4a2+11a+6
  5. 6x2+7x10
  6. 4x225x+6
  7. 24y235y+4
  8. 10y223y+12
  9. 14x211x+9
  10. 9x2+6x+8
  11. 4x228x+49
  12. 36x260x+25
  13. 27x26x8
  14. 24x2+17x20
  1. 6x2+23xy4y2
  2. 10x221xy27y2
  3. 8a2b218ab+9
  4. 12a2b2ab20
  5. 8u226uv+15v2
  6. 24m226mn+5n2
  7. 4a212ab+9b2
  8. 16a2+40ab+25b2
  9. 5(x+y)29(x+y)+4
  10. 7(xy)2+15(xy)18
  11. 7x422x2+3
  12. 5x441x2+8
  1. 4y63y310
  2. 12y6+4y35
  3. 5a4b4a2b218
  4. 21a4b4+5a2b24
  5. 6x6y6+17x3y3+10
  6. 16x6y6+46x3y3+15
  7. 8x2n10xn25
  8. 30x2n11xn6
  9. 36x2n+12axn+a2
  10. 9x2n12axn+4a2
  11. 3x2+14x+5
  12. 2x2+13x20
  1. x210x+24
  2. x2+8x+48
  3. 5412x2x2
  4. 60+5x5x2
  5. 4x3+16x2+20x
  6. 2x412x3+14x2
  7. 2x38x2y24xy2
  8. 6x39x2y6xy2
  9. 4a3b4a2b224ab3
  10. 15a4b33a3b2+6a2b3
  11. 3x5y+30x3y3+75xy5
  12. 45x5y260x3y4+20xy6
Answers to odd exercises:
201. (3x1)(x+7)
203. (2a+3)(3a+2)
205. (6x5)(x+2)
207. (8y1)(3y4)
209. Prime
211. (2x7)2
213. (9x+4)(3x2)
215. (6xy)(x+4y)
217. (4ab3)(2ab3)
219. (2u5v)(4u3v)
221. (2a3b)2
223. (x+y1)(5x+5y4)
225. (x23)(7x21)
227. (y32)(4y3+5)
229. (a2b22)(5a2b2+9)
231. (6x3y3+5)(x3y3+2)
233. (2xn5)(4xn+5)
235. (6xn+a)2
237. (x5)(3x+1)
239. (x2)(x+12)
241. 2(x3)(x+9)
243. 4x(x2+4x+5)
245. 2x(x+2y)(x6y)
247. 4ab(a3b)(a+2b)
249. 3xy(x2+5y2)2

F: Factor Trinomials.

Exercise 0.5.F 

 Factor the following trinomials.

  1. x28x48
  2. x215x+54
  3. x24x6
  4. x212xy+36y2
  5. x2+20xy+75y2
  6. x2+5x+150
  7. 2y2+20y+48
  8. 28x2+20x+3
  9. 150x2100x+6
  10. 24a238ab+3b2
  11. 27u23uv4v2
  12. 16x2y278xy+27
  13. 16m2+72mn+81n2
  1. 4x25x+20
  2. 25x435x2+6
  3. 2x4+7x2+3
  4. x6+3x3y310y6
  5. a68a3b3+15b6
  6. x2n2xn+1
  7. 6x2nxn2
  8. 25x2+30x+9
  9. 36a284ab+49b2
  10. 40x2+360x+810
  11. 5k370k2+245k
  12. 75u430u3v+3u2v2
  13. k216k+60
  1. m2+3m54
  2. x23x10
  3. x2+12xy+35y2
  4. r2+3rs28s2
  5. a2+4ab21b2
  6. p25pq36q2
  7. m25mn+30n2
  8. x3+5x224x
  9. 3y321y2+30y
  10. 5x4+10x375x2
  11. 5y2+14y+9
  12. 8x2+25x+3
  13. 10y253y11
  1. 6p219pq+10q2
  2. 81a2+153a+18
  3. 2x2+9x+4
  4. 18a29a+1
  5. 15p2+2p8
  6. 15x2+6x2
  7. 8a2+32a+24
  8. 3x2+3x36
  9. 48y2+12y36
  10. 18a257a21
  11. 3n412n396n2
Answers to odd exercises:
251. (x12)(x+4)
253. Prime
255. (x+5y)(x+15y)
257. 2(y12)(y+2)
259. 2(15x1)(5x3)
261. (3u+v)(9u4v)
263. (4m+9n)2
265. (5x26)(5x21)
267. (x3+5y3)(x32y3)
269. (xn1)2
271. (5x+3)2
273. 10(2x+9)2
275. 3u2(5uv)2
277. (m+9)(m6)
279. (x+5y)(x+7y)
281. (a+7b)(a3b)
283. Prime
285. 3y(y5)(y2)
287. (5y+9)(y+1)
289. (5y+1)(2y11)
291. 9(9a1)(a+2)
293. (3a1)(6a1)
295. Prime
297. 3(x+4)(x3)
299. 3(2a7)(3a+1)

G: Mixed Factoring Practice.

Exercise 0.5.G 

 Mixed practice. Factor completely.

  1. 425x2
  2. 8x3y3
  3. 9x212xy+4y2
  4. 30a257ab6b2
  5. 10a25a6ab+3b
  6. 3x34x2+9x12
  7. x2+4y2
  8. x2x+2
  9. 15a3b2+6a2b33ab4
  10. 54x263x
  11. 45x320x
  12. 12x470x3+50x2
  13. 20x2+32x3
  14. x3y+9xy3
  15. 24a4b2+3ab5
  16. 64a6b61
  17. 64x2+1
  1. x ^ { 3 } + x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 }
  2. 8y^3+16y^2−24y
  3. 5y^3−15y^2−270y
  4. 16x^3−36x
  5. 27y^2−48
  6. 4x^2+20xy+25y^2
  7. 9x^2−24xy+16y^2
  8. 50x^3y+72xy
  9. 250m^3+432n^3
  10. 4a^5b−64ab
  11. 7xy^5−7xy
  12. 6x^2-12xc+6bx-12bc
  13. 16x^2+24xy−4x−6y
  14. 4p^2q−16pq+12q
  15. 6pq^2−9pq−6p
  16. 4x^2-12xy+9y^2-25
  17. 16x^2-24xy+9y^2-64
  1. 80a^2+120a^3
  2. 5m(m−1)+3(m−1)
  3. x^2+13x+36
  4. p^2+pq−12q^2
  5. xy−8y+7x−56
  6. 40r^2+810
  7. 9s^2−12s+4
  8. 6x^2−11x−10
  9. 3x^2−75y^2
  10. 6u^2+3u−18
  11. x^3+125
  12. 32x^5y^2−162xy^2
  13. 6x^4−19x^2+15
  14. 3x^3−36x^2+108x
  15. 24x^3+44x^2
  16. 24a^4−9a^3
  17. 16n^2−56mn+49m^2
  1. 6a^2−25a−9
  2. 5u^4−45u^2
  3. n^4−81
  4. 64j^2+225
  5. 5x^2+5x−60
  6. b^3−64
  7. m^3+125
  8. 2b^2−2bc+5cb−5c^2
  9. 48x^5y^2−243xy^2
  10. 5q^2−15q−90
  11. 4u^5v+4u^2v^3
  12. 10m^4−6250
  13. 60x^2y−75xy+30y
  14.  16x^2 + 32x + 16 - 121
  15.  9x^2 + 42x + 49 - 9
  16.  16x^2 + 72x + 81 - 144x^2
  17.  9x^2 + 12x + 4 - 36x^2
Answers to odd exercises:
301. ( 2 - 5 x ) ( 2 + 5 x )
303. ( 3 x - 2 y ) ^ { 2 }
305. ( 2 a - 1 ) ( 5 a - 3 b )
307. Prime
309. 3 a b ^ { 2 } \left( 5 a ^ { 2 } + 2 a b - b ^ { 2 } \right)
311. 5 x ( 3 x + 2 ) ( 3 x - 2 )
313. - ( 10 x - 1 ) ( 2 x - 3 )
315. 3 a b ^ { 2 } ( 2 a + b ) \left( 4 a ^ { 2 } - 2 a b + b ^ { 2 } \right)
317. Prime
319. 8y(y−1)(y+3)
321. 4x(2x−3)(2x+3)
323. (2x+5y)^2
325. 2xy(25x^2+36)
327. 4ab(a^2+4)(a−2)(a+2)
329. 6(x+b)(x−2c)
331. 4q(p−3)(p−1)
333. (2x−3y−5)(2x−3y+5)
335. 40a^2(2+3a)
337. (x+9)(x+4)
339. (x−8)(y+7)
341. (3s−2)^2
343. 3(x+5y)(x−5y)
345. (x+5)(x^2−5x+25)
347. (3x^2−5)(2x^2−3)
349. 4x^2(6x+11)
351. (4n−7m)^2
353. 5u^2(u+3)(u−3)
355. prime
357. (b−4)(b^2+4b+16)
359. (2b+5c)(b−c)
361. 5(q+3)(q−6)
363. 10(m−5)(m+5)(m^2+25)
365.  (4x -7)(4x+15)  
367.   (-8x +9)(16x +9) 

.


0.05e: Exercises - Factoring is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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