Algebra: Factoring of Polynomials
For convenience and quick browsing, the problems are grouped into polynomials that are:
- product of a monomial and a polynomial
- grouped into terms containing a common factor
- difference of two squares
- difference of two cubes
- sum of two cubes
- product of two binomials
The last group of polynomials given above consists of polynomials that are:
- product of two binomials having a common factor
- product of any two binomials
- square of a binomial sum
- square of a binomial difference
Some of these grouped polynomials contain factors which are a combination of a monomial, a binomial, and a trinomial. Polynomials that are:
- difference of two squares can be the products of two binomials, or products of two trinomials, or products of a monomial and two binomials.
- sum of two cubes can be products of a binomial and a trinomial, or products of a monomial, a binomial, and a trinomial
Polynomials With Monomial As One of Its Factors
| 1. | 15x4 + 15x3y |
| Answer: (3x3)(5x + 5y) |
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| 2. | 4x3 - 4x2y4 |
| Answer: (4x2)(x - y4) |
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| 3. | 5x5 - 5x3y2 |
| Answer: (x3)(5x2 - 5y2) |
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| 4. | 63s6t3 - 42s3t3u + 126s3t3 |
| Answer: (21s3t3)(3s3 - 2u + 6) |
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| 5. | 14x5 + 6x2y |
| Answer: (2x2)(7x3 + 3y) |
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| 6. | 18x6y3 - 15x2y3z2 + 3x2y3 |
| Answer: (3x2y3)(6x4 - 5z2 + 1) |
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| 7. | 14a4b4 + 56a3b4 |
| Answer: (14a3b4)(a + 4) |
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| 8. | 7x5 + xy2 |
| Answer: (x)(7x4 + y2) |
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| 9. | 7x4 + 21x3y3 |
| Answer: (7x3)(x + 3y3) |
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| 10. | 24x6y4 - 192x3y4 |
| Answer: (24x3y4)(x3 - 8) |
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| 11. | 35x3 - 49xy2 |
| Answer: (7x)(5x2 - 7y2) |
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| 12. | 48a4b2 + 12a3b2 |
| Answer: (12a3b2)(4a + 1) |
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| 13. | 56x4 - 40xy4 |
| Answer: (8x)(7x3 - 5y4) |
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| 14. | 20a3b2 + 60ab2 |
| Answer: (20ab2)(a2 + 3) |
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| 15. | 120s5t3 - 75s2t3u2 + 30s2t3 |
| Answer: (15s2t3)(8s3 - 5u2 + 2) |
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| 16. | 24x4y - 6x3y |
| Answer: (6x3y)(4x - 1) |
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| 17. | 105x5y - 35x3y |
| Answer: (35x3y)(3x2 - 1) |
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| 18. | 30x5 - 15x2y3 |
| Answer: (5x2)(6x3 - 3y3) |
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| 19. | 6x4 + 8x3y4 |
| Answer: (x3)(6x + 8y4) |
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| 20. | 20x4y3 - 40xy3 |
| Answer: (20xy3)(x3 - 2) |
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Polynomials That Can Be Grouped Into Terms Having A Common Factor
| 1. | su4 + sv + tu4 + tv |
| Answer: (s + t)(u4 + v) |
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| 2. | wy + 3wz + xy + 3xz |
| Answer: (w + x)(y + 3z) |
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| 3. | 24su - 42sv + 20tu - 35tv |
| Answer: (6s + 5t)(4u - 7v) |
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| 4. | 20w2y4 + 35w2z4 + 8xy4 + 14xz4 |
| Answer: (5w2 + 2x)(4y4 + 7z4) |
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| 5. | 64w4y3 - 40w4z2 + 8xy3 - 5xz2 |
| Answer: (8w4 + x)(8y3 - 5z2) |
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| 6. | 3ab + 3a + b2 + b |
| Answer: (3a + b)(b + 1) |
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| 7. | 5xy - 20x + 2y2 - 8y |
| Answer: (5x + 2y)(y - 4) |
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| 8. | 28s3t - 35s3 + 24t2 - 30t |
| Answer: (7s3 + 6t)(4t - 5) |
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| 9. | 7su + sv - 7tu - tv |
| Answer: (s - t)(7u + v) |
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| 10. | su3 + sv4 + tu3 + tv4 |
| Answer: (s + t)(u3 + v4) |
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| 11. | 8wy + 32wz + 3xy + 12xz |
| Answer: (8w + 3x)(y + 4z) |
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| 12. | 6su - 16sv + 3tu - 8tv |
| Answer: (2s + t)(3u - 8v) |
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| 13. | 12w2y + 15w2z2 + 8x3y + 10x3z2 |
| Answer: (3w2 + 2x3)(4y + 5z2) |
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| 14. | 3w3y4 - 8w3z3 + 15x4y4 - 40x4z3 |
| Answer: (w3 + 5x4)(3y4 - 8z3) |
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| 15. | 4ab + 2a + 10b2 + 5b |
| Answer: (2a + 5b)(2b + 1) |
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| 16. | 28xy - 4x + 7y2 - y |
| Answer: (4x + y)(7y - 1) |
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| 17. | 24st2 - 15s + 64t4 - 40t2 |
| Answer: (3s + 8t2)(8t2 - 5) |
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| 18. | 5su + 2sv - 5tu - 2tv |
| Answer: (s - t)(5u + 2v) |
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| 19. | 12ab + 42a + 14b2 + 49b |
| Answer: (6a + 7b)(2b + 7) |
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| 20. | 6xy - 10x + 9y2 - 15y |
| Answer: (2x + 3y)(3y - 5) |
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Polynomials That Are The Difference of Two Squares
You will find that some of the problems given below can be factored twice, that is, after factoring an expression one of the resulting factors is still factorable.
Also, in some problems a group of terms is treated as a single term.
| 1. | x2 - 25 |
| Answer: (x - 5)(x + 5) |
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| 2. | 36x2 - 25 |
| Answer: (6x - 5)(6x + 5) |
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| 3. | 25x2 - 49 |
| Answer: (5x - 7)(5x + 7) |
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| 4. | x2 - 9 |
| Answer: (x - 3)(x + 3) |
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| 5. | 16x2 - 25 |
| Answer: (4x - 5)(4x + 5) |
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| 6. | x2 - (y - z)2 |
| Answer: [ x + (y - z) ] [ (x - (y - z) ] |
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| 7. | (x + 8)2 - 9 |
| Answer: [ (x + 8) + 3 ] [ (x + 8) - 3 ] |
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| 8. | (x + 3)2 - 1 |
| Answer: [ (x + 3) + 1 ] [ (x + 3) - 1 ] |
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| 9. | 49x2 - 9(y - z)2 |
| Answer: [ 7x + 3(y - z) ] [ (7x - 3(y - z) ] |
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| 10. | (x + 5)2 - 9 |
| Answer: [ (x + 5) + 3 ] [ (x + 5) - 3 ] |
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| 11. | (x + 1)2 - 36 |
| Answer: [ (x + 1) + 6 ] [ (x + 1) - 6 ] |
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| 12. | (x + 8)2 - 25 |
| Answer: [ (x + 8) + 5 ] [ (x + 8) - 5 ] |
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| 13. | 16x2 - 49(y - z)2 |
| Answer: [ 4x + 7(y - z) ] [ (4x - 7(y - z) ] |
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| 14. | 49x4y2 - 49x4z2 |
| Answer: 49x4 (y + z) (y - z) |
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| 15. | 49x2y2 - 441x2z2 |
| Answer: 49x2 (y + 3z) (y - 3z) |
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| 16. | 9x2y2 - 36x2z2 |
| Answer: 9x2 (y + 2z) (y - 2z) |
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| 17. | 16x2y2 - 144x2z2 |
| Answer: 16x2 (y + 3z) (y - 3z) |
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| 18. | 4x2y2 - 100x2z2 |
| Answer: 4x2 (y + 5z) (y - 5z) |
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| 19. | x4 - 256y4 |
| Answer: (x + 4y) (x - 4y) (x2 + 16y2) |
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| 20. | x4 - 81y4 |
| Answer: (x + 3y) (x - 3y) (x2 + 9y2) |
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Polynomials That Are The Difference of Two Cubes
| 1. | 27x3y6 - 125z3 |
| Answer: (3xy2 - 5z) (9x2y4 + 15xy2z + 25z2) |
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| 2. | 343y6 - 512 |
| Answer: (7y2 - 8) (49y4 + 56y2 + 64) |
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| 3. | 8a6 - 125b3 |
| Answer: (2a2 - 5b) (4a4 + 10a2b + 25b2) |
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| 4. | x3 - 8 |
| Answer: (x - 2) (x2 + 2x + 4) |
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| 5. | 8a3 - 512b3 |
| Answer: (2a - 8b) (4a2 + 16ab + 64b2) |
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| 6. | 343x3y6 - 64z3 |
| Answer: (7xy2 - 4z) (49x2y4 + 28xy2z + 16z2) |
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| 7. | 512y3 - 125 |
| Answer: (8y - 5) (64y2 + 40y + 25) |
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| 8. | 27a6 - 125b3 |
| Answer: (3a2 - 5b) (9a4 + 15a2b + 25b2) |
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| 9. | 8a3 - 125b3 |
| Answer: (2a - 5b) (4a2 + 10ab + 25b2) |
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| 10. | 125y6 - 216 |
| Answer: (5y2 - 6) (25y4 + 30y2 + 36) |
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| 11. | 343y3 - 216 |
| Answer: (7y - 6) (49y2 + 42y + 36) |
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| 12. | x6 - 8 |
| Answer: (x2 - 2) (x4 + 2x2 + 4) |
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| 13. | 8y3 - 125 |
| Answer: (2y - 5) (4y2 + 10y + 25) |
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| 14. | 8y3 - 216 |
| Answer: (2y - 6) (4y2 + 12y + 36) |
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| 15. | 343y6 - 27 |
| Answer: (7y2 - 3) (49y4 + 21y2 + 9) |
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| 16. | x6 - 64 |
| Answer: (x2 - 4) (x4 + 4x2 + 16) |
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| 17. | x6 - 27 |
| Answer: (x2 - 3) (x4 + 3x2 + 9) |
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| 18. | x3 - 27 |
| Answer: (x - 3) (x2 + 3x + 9) |
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| 19. | 8y6 - 343 |
| Answer: (2y2 - 7) (4y4 + 14y2 + 49) |
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| 20. | 27y6 - 27 |
| Answer: (3y2 - 3) (9y4 + 9y2 + 9) |
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| 21. | 512a3 - 8b3 |
| Answer: (8a - 2b) (64a2 + 16ab + 4b2) |
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| 22. | 64a6 - 343b3 |
| Answer: (4a2 - 7b) (16a4 + 28a2b + 49b2) |
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| 23. | 8a3 - 8b3 |
| Answer: (2a - 2b) (4a2 + 4ab + 4b2) |
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| 24. | x3 - 125 |
| Answer: (x - 5) (x2 + 5x + 25) |
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| 25. | 512a6 - 8b3 |
| Answer: (8a2 - 2b) (64a4 + 16a2b + 4b2) |
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Polynomials That Are The Sum of Two Cubes
| 1. | 512x6 + 27y6 |
| Answer: (8x2 + 3y2) (64x4 - 24x2y2 + 9y4) |
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| 2. | 8x6 + 27y6 |
| Answer: (2x2 + 3y2) (4x4 - 6x2y2 + 9y4) |
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| 3. | 8a6 + 125b6 |
| Answer: (2a2 + 5b2) (4a4 - 10a2b2 + 25b4) |
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| 4. | 512x3 + 8y3 |
| Answer: (8x + 2y) (64x2 - 16xy + 4y2) |
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| 5. | 8x6 + 64y6 |
| Answer: (2x2 + 4y2) (4x4 - 8x2y2 + 16y4) |
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| 6. | 8m3 + 343n3 |
| Answer: (2m + 7n) (4m2 - 14mn + 49n2) |
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| 7. | 64s3x + 8t3x |
| Answer: x (4s + 2t)(16s2 - 8st + 4t2) |
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| 8. | 8x3 + 216y3 |
| Answer: (2x + 6y) (4x2 - 12xy + 36y2) |
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| 9. | 125x6y + 343y |
| Answer: y (5x2 + 7)(25x4 - 35x2 + 49) |
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| 10. | 125x3 + 8y3 |
| Answer: (5x + 2y) (25x2 - 10xy + 4y2) |
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| 11. | 512x6 + 343y6 |
| Answer: (8x2 + 7y2) (64x4 - 56x2y2 + 49y4) |
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| 12. | 8x3 + 125y3 |
| Answer: (2x + 5y) (4x2 - 10xy + 25y2) |
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| 13. | 125a6x + 64b6x |
| Answer: x (5a2 + 4b2)(25a4 - 20a2b2 + 16b4) |
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| 14. | 64m3y + 27n3y |
| Answer: y (4m + 3n)(16m2 - 12mn + 9n2) |
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| 15. | 64s3x + 125t3x |
| Answer: x (4s + 5t)(16s2 - 20st + 25t2) |
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| 16. | 216x3z + 343y3z |
| Answer: z (6x + 7y)(36x2 - 42xy + 49y2) |
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| 17. | 8s6x + 125t6x |
| Answer: x (2s2 + 5t2)(4s4 - 10s2t2 + 25t4) |
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| 18. | 343a3c + 64b3c |
| Answer: c (7a + 4b)(49a2 - 28ab + 16b2) |
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| 19. | 512s6x + 343t6x |
| Answer: x (8s2 + 7t2)(64s4 - 56s2t2 + 49t4) |
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| 20. | 8s6u + 27t6u |
| Answer: u (2s2 + 3t2)(4s4 - 6s2t2 + 9t4) |
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The following problems are about polynomials which are the products of two binomials.
Polynomials That Are The Products of Two Binomials Having a Common Factor
| 1. | x2 + 11x + 30 |
| Answer: (x + 5)(x + 6) |
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| 2. | x2 - 11x + 24 |
| Answer: (x - 3)(x - 8) |
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| 3. | x2 + 8x + 15 |
| Answer: (x + 3)(x + 5) |
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| 4. | 49x2 - 35x + 4 |
| Answer: (7x - 1)(7x - 4) |
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| 5. | x2 - 9x + 14 |
| Answer: (x - 2)(x - 7) |
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| 6. | x2 + 5x + 4 |
| Answer: (x + 1)(x + 4) |
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| 7. | x2 - x - 30 |
| Answer: (x - 6)(x + 5) |
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| 8. | x2 + 7x + 10 |
| Answer: (x + 5)(x + 2) |
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| 9. | 25x2 + 65x + 42 |
| Answer: (5x + 6)(5x + 7) |
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| 10. | x2 + x - 56 |
| Answer: (x - 7)(x + 8) |
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| 11. | x2 - 9x + 8 |
| Answer: (x - 1)(x - 8) |
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| 12. | x2 + 2x - 15 |
| Answer: (x - 3)(x + 5) |
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| 13. | x2 + 13x + 42 |
| Answer: (x + 6)(x + 7) |
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| 14. | x2 + 11x + 24 |
| Answer: (x + 8)(x + 3) |
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| 15. | x2 - 11x + 30 |
| Answer: (x - 5)(x - 6) |
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| 16. | x2 - 13x + 40 |
| Answer: (x - 5)(x - 8) |
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| 17. | x2 + x - 30 |
| Answer: (x - 5)(x + 6) |
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| 18. | x2 + 3x + 2 |
| Answer: (x + 2)(x + 1) |
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| 19. | x2 - 2x - 3 |
| Answer: (x - 3)(x + 1) |
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| 20. | x2 - 8x + 15 |
| Answer: (x - 5)(x - 3) |
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Polynomials That Are The Products of Any Two Binomials
| 1. | 24x2 + 35x - 49 |
| Answer: (8x - 7)(3x + 7) |
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| 2. | 2x2 + 7x - 4 |
| Answer: (2x - 1)(x + 4) |
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| 3. | 24x2 + 34x + 7 |
| Answer: (6x + 7)(4x + 1) |
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| 4. | 32x2 + 28x + 3 |
| Answer: (8x + 1)(4x + 3) |
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| 5. | 33x2 + 14x + 1 |
| Answer: (3x + 1)(11x + 1) |
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| 6. | 12x2 + 11x - 5 |
| Answer: (3x - 1)(4x + 5) |
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| 7. | 5x2 + 33x - 56 |
| Answer: (5x - 7)(x + 8) |
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| 8. | 8x2 - 9x + 1 |
| Answer: (x - 1)(8x - 1) |
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| 9. | 3x2 + 19x + 6 |
| Answer: (x + 6)(3x + 1) |
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| 10. | 6x2 - 25x + 14 |
| Answer: (3x - 2)(2x - 7) |
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| 11. | 35x2 + 17x - 30 |
| Answer: (7x - 5)(5x + 6) |
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| 12. | 7x2 + 38x + 15 |
| Answer: (7x + 3)(x + 5) |
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| 13. | 3x2 + 32x + 64 |
| Answer: (x + 8)(3x + 8) |
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| 14. | 3x2 - 13x + 12 |
| Answer: (x - 3)(3x - 4) |
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| 15. | 10x2 + 21x + 9 |
| Answer: (2x + 3)(5x + 3) |
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| 16. | 6x2 - 23x + 20 |
| Answer: (3x - 4)(2x - 5) |
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| 17. | 10x2 - 47x + 42 |
| Answer: (2x - 7)(5x - 6) |
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| 18. | 6x2 - 43x + 42 |
| Answer: (6x - 7)(x - 6) |
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| 19. | 15x2 + 4x - 32 |
| Answer: (3x - 4)(5x + 8) |
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| 20. | 3x2 - 14x - 5 |
| Answer: (x - 5)(3x + 1) |
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Polynomials That Are The Square of a Binomial Sum
| 1. | 169x2 + 364x + 196 |
| Answer: (13x + 14)(13x + 14) |
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| 2. | 64x2 + 80x + 25 |
| Answer: (8x + 5)(8x + 5) |
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| 3. | 25x2 + 40x + 16 |
| Answer: (5x + 4)(5x + 4) |
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| 4. | x2 + 10x + 25 |
| Answer: (x + 5)(x + 5) |
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| 5. | 64x2 + 48x + 9 |
| Answer: (8x + 3)(8x + 3) |
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| 6. | 25x2 + 60x + 36 |
| Answer: (5x + 6)(5x + 6) |
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| 7. | 16x2 + 8x + 1 |
| Answer: (4x + 1)(4x + 1) |
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| 8. | x2 + 22x + 121 |
| Answer: (x + 11)(x + 11) |
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| 9. | 169x2 + 78x + 9 |
| Answer: (13x + 3)(13x + 3) |
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| 10. | x2 + 4x + 4 |
| Answer: (x + 2)(x + 2) |
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| 11. | 49x2 + 56x + 16 |
| Answer: (7x + 4)(7x + 4) |
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| 12. | x2 + 2x + 1 |
| Answer: (x + 1)(x + 1) |
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| 13. | x2 + 14x + 49 |
| Answer: (x + 7)(x + 7) |
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| 14. | 169x2 + 156x + 36 |
| Answer: (13x + 6)(13x + 6) |
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| 15. | x2 + 16x + 64 |
| Answer: (x + 8)(x + 8) |
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| 16. | x2 + 26x + 169 |
| Answer: (x + 13)(x + 13) |
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| 17. | 64x2 + 16x + 1 |
| Answer: (8x + 1)(8x + 1) |
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| 18. | x2 + 8x + 16 |
| Answer: (x + 4)(x + 4) |
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| 19. | 25x2 + 110x + 121 |
| Answer: (5x + 11)(5x + 11) |
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| 20. | 49x2 + 42x + 9 |
| Answer: (7x + 3)(7x + 3) |
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Polynomials That Are The Square of a Binomial Difference
| 1. | 9x2 - 30x + 25 |
| Answer: (3x - 5)(3x - 5) |
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| 2. | 169x2 - 364x + 196 |
| Answer: (13x - 14)(13x - 14) |
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| 3. | 25x2 - 40x + 16 |
| Answer: (5x - 4)(5x - 4) |
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| 4. | 49x2 - 42x + 9 |
| Answer: (7x - 3)(7x - 3) |
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| 5. | 9x2 - 60x + 100 |
| Answer: (3x - 10)(3x - 10) |
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| 6. | 64x2 - 80x + 25 |
| Answer: (8x - 5)(8x - 5) |
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| 7. | 36x2 - 84x + 49 |
| Answer: (6x - 7)(6x - 7) |
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| 8. | x2 - 26x + 169 |
| Answer: (x - 13)(x - 13) |
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| 9. | 9x2 - 12x + 4 |
| Answer: (3x - 2)(3x - 2) |
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| 10. | x2 - 2x + 1 |
| Answer: (x - 1)(x - 1) |
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| 11. | x2 - 14x + 49 |
| Answer: (x - 7)(x - 7) |
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| 12. | x2 - 4x + 4 |
| Answer: (x - 2)(x - 2) |
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| 13. | 4x2 - 20x + 25 |
| Answer: (2x - 5)(2x - 5) |
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| 14. | x2 - 8x + 16 |
| Answer: (x - 4)(x - 4) |
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| 15. | 25x2 - 80x + 64 |
| Answer: (5x - 8)(5x - 8) |
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| 16. | x2 - 16x + 64 |
| Answer: (x - 8)(x - 8) |
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| 17. | x2 - 6x + 9 |
| Answer: (x - 3)(x - 3) |
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| 18. | 64x2 - 16x + 1 |
| Answer: (8x - 1)(8x - 1) |
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| 19. | x2 - 10x + 25 |
| Answer: (x - 5)(x - 5) |
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| 20. | 49x2 - 70x + 25 |
| Answer: (7x - 5)(7x - 5) |
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