Factoring Polynomials Worksheet 2 displaying difference and sum of squares formulas and 30 algebraic expressions for practice.

I'll work through each polynomial factoring problem step by step, then provide all the final answers.

Let me solve these systematically:

Problem 1: 3p² - 2p - 5
Looking for factors of (3)(-5) = -15 that add to -2: -5 and +3
3p² - 5p + 3p - 5 = p(3p - 5) + 1(3p - 5) = (p + 1)(3p - 5)

Problem 2: 3x² - 6x - 45
Factor out 3: 3(x² - 2x - 15)
Factors of -15 that add to -2: -5 and +3
= 3(x - 5)(x + 3)

Problem 3: 4w² - 15w - 25
Looking for factors of (4)(-25) = -100 that add to -15: -20 and +5
4w² - 20w + 5w - 25 = 4w(w - 5) + 5(w - 5) = (4w + 5)(w - 5)

Problem 4: 4y² - 20y + 25
Perfect square trinomial: (2y)² - 2(2y)(5) + 5² = (2y - 5)²

Problem 5: 10j³ - 130j² + 400j
Factor out 10j: 10j(j² - 13j + 40)
Factors of 40 that add to -13: -8 and -5
= 10j(j - 8)(j - 5)

Problem 6: -49 + 35b - 4b²
Rearrange: -4b² + 35b - 49
Factor out -1: -(4b² - 35b + 49)
Looking for factors of (4)(49) = 196 that add to -35: -28 and -7
-(4b² - 28b - 7b + 49) = -(4b(b - 7) - 7(b - 7)) = -(4b - 7)(b - 7)
Or: (7 - 4b)(b - 7)

Problem 7: 2n² + 5mn + 2m²
Looking for factors of (2)(2) = 4 that add to 5: 4 and 1
2n² + 4mn + mn + 2m² = 2n(n + 2m) + m(n + 2m) = (2n + m)(n + 2m)

Problem 8: 4y² - 17y + 4
Looking for factors of (4)(4) = 16 that add to -17: -16 and -1
4y² - 16y - y + 4 = 4y(y - 4) - 1(y - 4) = (4y - 1)(y - 4)

Problem 9: -5x² + 18x - 9
Factor out -1: -(5x² - 18x + 9)
Looking for factors of (5)(9) = 45 that add to -18: -15 and -3
-(5x² - 15x - 3x + 9) = -(5x(x - 3) - 3(x - 3)) = -(5x - 3)(x - 3)
Or: (3 - 5x)(x - 3)

Problem 10: 12u² - 4u - 16
Factor out 4: 4(3u² - u - 4)
Looking for factors of (3)(-4) = -12 that add to -1: -4 and +3
4(3u² - 4u + 3u - 4) = 4(u(3u - 4) + 1(3u - 4)) = 4(u + 1)(3u - 4)

Problem 11: z⁴ + 6z² - 7z²
Simplify first: z⁴ - z² = z²(z² - 1) = z²(z - 1)(z + 1)

Problem 12: 9k² + 66k + 21
Factor out 3: 3(3k² + 22k + 7)
Looking for factors of (3)(7) = 21 that add to 22: 21 and 1
3(3k² + 21k + k + 7) = 3(3k(k + 7) + 1(k + 7)) = 3(3k + 1)(k + 7)

Problem 13: 6x² - 7xy - 3y²
Looking for factors of (6)(-3) = -18 that add to -7: -9 and +2
6x² - 9xy + 2xy - 3y² = 3x(2x - 3y) + y(2x - 3y) = (3x + y)(2x - 3y)

Problem 14: 6b² - 15b - 9
Factor out 3: 3(2b² - 5b - 3)
Looking for factors of (2)(-3) = -6 that add to -5: -6 and +1
3(2b² - 6b + b - 3) = 3(2b(b - 3) + 1(b - 3)) = 3(2b + 1)(b - 3)

Problem 15: 4w² - 17w - 15
Looking for factors of (4)(-15) = -60 that add to -17: -20 and +3
4w² - 20w + 3w - 15 = 4w(w - 5) + 3(w - 5) = (4w + 3)(w - 5)

Problem 16: 4y² - 17y + 15
Looking for factors of (4)(15) = 60 that add to -17: -12 and -5
4y² - 12y - 5y + 15 = 4y(y - 3) - 5(y - 3) = (4y - 5)(y - 3)

Problem 17: 15m³ + 9m² - 6m
Factor out 3m: 3m(5m² + 3m - 2)
Looking for factors of (5)(-2) = -10 that add to 3: 5 and -2
3m(5m² + 5m - 2m - 2) = 3m(5m(m + 1) - 2(m + 1)) = 3m(5m - 2)(m + 1)

Problem 18: z⁴ + 2z² - 1
This is a perfect square: (z²)² + 2(z²)(1) - 1... wait, let me check
Actually: z⁴ + 2z² - 1 doesn't factor nicely over integers. Let me reconsider.
If we substitute u = z²: u² + 2u - 1. Using quadratic formula: u = (-2 ± (4+4))/2 = (-2 ± 2√2)/2 = -1 ± √2
So z² = -1 + √2 or z² = -1 - 2 (not real)
This doesn't factor over integers. Let me check if there's an error in the problem or if it should be z⁴ + 2z² + 1 = (z² + 1)²

Wait, looking at the pattern, this might be meant to be z⁴ + 2z² + 1 = (z² + 1)², but as written it's z⁴ + 2z² - 1 which doesn't have integer factors.

Problem 19: h³ - k³
Difference of cubes: (h - k)(h² + hk + k²)

Problem 20: 8p³ + r³
Sum of cubes: (2p)³ + r³ = (2p + r)((2p)² - (2p)(r) + r²) = (2p + r)(4p² - 2pr + r²)

Problem 21: 125h³ - 27k³
Difference of cubes: (5h)³ - (3k)³ = (5h - 3k)((5h)² + (5h)(3k) + (3k)²) = (5h - 3k)(25h² + 15hk + 9k²)

Problem 22: 27c - 8d³
Difference of cubes: (3c²)³ - (2d)³ = (3c² - 2d)((3c²)² + (3c²)(2d) + (2d)²) = (3c² - 2d)(9c⁴ + 6c²d + 4d²)

Problem 23: 250a³ + 2b³
Factor out 2: 2(125a³ + b³) = 2((5a)³ + b³) = 2(5a + b)((5a)² - (5a)(b) + b²) = 2(5a + b)(25a² - 5ab + b²)

Problem 24: 36w² - 49z²
Difference of squares: (6w)² - (7z)² = (6w - 7z)(6w + 7z)

Problem 25: 5x² - 45
Factor out 5: 5(x² - 9) = 5(x - 3)(x + 3)

Problem 26: a⁵x - 16ax³
Factor out ax: ax(a⁴ - 16x²) = ax((a²)² - (4x)²) = ax(a² - 4x)(a² + 4x)
Further: ax(a - 2√x)(a + 2√x)(a² + 4x)... but this introduces radicals
Better: ax(a² - 4x)(a² + 4x) where a² - 4x can be written as (a - 2√x)(a + 2√x) only if we allow radicals
Actually, if we treat this as difference of squares in a different way:
ax(a⁴ - 16x²) = ax(a² - 4x)(a² + 4x)
But a² - 4x doesn't factor further over integers unless x is a perfect square times something.
The answer is: ax(a² - 4x)(a² + 4x)

Problem 27: 16a⁴ - b⁴
Difference of squares: (4a²)² - (b²)² = (4a² - b²)(4a² + b²)
Further: ((2a)² - b²)(4a² + b²) = (2a - b)(2a + b)(4a² + b²)

Problem 28: 3x⁷ + 36x⁵ + 108x³
Factor out 3x³: 3x³(x⁴ + 12x² + 36)
Perfect square: 3x³((x²)² + 2(x²)(6) + 6²) = 3x³(x² + 6)²

Problem 29: 4u² + 20u + 25
Perfect square trinomial: (2u)² + 2(2u)(5) + 5² = (2u + 5)²

Problem 30: 121b² - 66bd + 9d²
Perfect square trinomial: (11b)² - 2(11b)(3d) + (3d)² = (11b - 3d)²

Final Answer:
1. (p + 1)(3p - 5)
2. 3(x - 5)(x + 3)
3. (4w + 5)(w - 5)
4. (2y - 5)²
5. 10j(j - 8)(j - 5)
6. (7 - 4b)(b - 7) or -(4b - 7)(b - 7)
7. (2n + m)(n + 2m)
8. (4y - 1)(y - 4)
9. (3 - 5x)(x - 3) or -(5x - 3)(x - 3)
10. 4(u + 1)(3u - 4)
11. z²(z - 1)(z + 1)
12. 3(3k + 1)(k + 7)
13. (3x + y)(2x - 3y)
14. 3(2b + 1)(b - 3)
15. (4w + 3)(w - 5)
16. (4y - 5)(y - 3)
17. 3m(5m - 2)(m + 1)
18. Does not factor over integers (or possibly meant to be z⁴ + 2z² + 1 = (z² + 1)²)
19. (h - k)(h² + hk + k²)
20. (2p + r)(4p² - 2pr + r²)
21. (5h - 3k)(25h² + 15hk + 9k²)
22. (3c² - 2d)(9c + 6c²d + 4d²)
23. 2(5a + b)(25a² - 5ab + b²)
24. (6w - 7z)(6w + 7z)
25. 5(x - 3)(x + 3)
26. ax(a² - 4x)(a² + 4x)
27. (2a - b)(2a + b)(4a² + b²)
28. 3x³(x² + 6)²
29. (2u + 5)²
30. (11b - 3d)²