Factoring Polynomials Worksheet 2 with difference and sum of squares formulas and 30 algebraic expressions.

Problem Overview:

The worksheet provided involves factoring polynomials using various techniques, including the difference of squares and the sum of squares formulas. The goal is to factor each polynomial completely.

Key Formulas:

1. Difference of Squares:
\[
a^2 - b^2 = (a - b)(a + b)
\]
2. Sum of Squares:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
3. Difference of Cubes:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Additionally, other techniques such as factoring out the greatest common factor (GCF), grouping, and recognizing perfect square trinomials will be used where applicable.

Solution:

#### Section 1: Difference of Squares
We will solve problems that can be factored using the difference of squares formula.

---

#### Problem 1: \( 3p^2 - 2p - 5 \)
This is not a difference of squares. It is a quadratic trinomial. We factor it by finding two numbers that multiply to \( 3 \cdot (-5) = -15 \) and add to \(-2\). These numbers are \(-5\) and \(3\).

\[
3p^2 - 2p - 5 = 3p^2 - 5p + 3p - 5
\]
\[
= p(3p - 5) + 1(3p - 5)
\]
\[
= (3p - 5)(p + 1)
\]

Answer: \((3p - 5)(p + 1)\)

---

#### Problem 4: \( 4y^2 - 20y + 25 \)
This is a perfect square trinomial:
\[
4y^2 - 20y + 25 = (2y)^2 - 2 \cdot 2y \cdot 5 + 5^2 = (2y - 5)^2
\]

Answer: \((2y - 5)^2\)

---

#### Problem 5: \( 10j^2 - 130j^2 + 400j \)
First, simplify the expression:
\[
10j^2 - 130j^2 + 400j = -120j^2 + 400j
\]
Factor out the GCF, which is \(20j\):
\[
-120j^2 + 400j = 20j(-6j + 20) = 20j(20 - 6j)
\]

Answer: \(20j(20 - 6j)\)

---

#### Problem 8: \( 4y^2 - 17y + 4 \)
This is a quadratic trinomial. We factor it by finding two numbers that multiply to \( 4 \cdot 4 = 16 \) and add to \(-17\). These numbers are \(-16\) and \(-1\).

\[
4y^2 - 17y + 4 = 4y^2 - 16y - y + 4
\]
\[
= 4y(y - 4) - 1(y - 4)
\]
\[
= (4y - 1)(y - 4)
\]

Answer: \((4y - 1)(y - 4)\)

---

#### Problem 9: \( -5x^2 + 18x - 9 \)
Factor out the negative sign first:
\[
-5x^2 + 18x - 9 = -(5x^2 - 18x + 9)
\]
Now, factor \(5x^2 - 18x + 9\). Find two numbers that multiply to \(5 \cdot 9 = 45\) and add to \(-18\). These numbers are \(-15\) and \(-3\).

\[
5x^2 - 18x + 9 = 5x^2 - 15x - 3x + 9
\]
\[
= 5x(x - 3) - 3(x - 3)
\]
\[
= (5x - 3)(x - 3)
\]

Thus:
\[
-5x^2 + 18x - 9 = -(5x - 3)(x - 3)
\]

Answer: \(-(5x - 3)(x - 3)\)

---

#### Problem 10: \( 12u^2 - 4u - 16 \)
Factor out the GCF, which is \(4\):
\[
12u^2 - 4u - 16 = 4(3u^2 - u - 4)
\]
Now, factor \(3u^2 - u - 4\). Find two numbers that multiply to \(3 \cdot (-4) = -12\) and add to \(-1\). These numbers are \(-4\) and \(3\).

\[
3u^2 - u - 4 = 3u^2 - 4u + 3u - 4
\]
\[
= u(3u - 4) + 1(3u - 4)
\]
\[
= (3u - 4)(u + 1)
\]

Thus:
\[
12u^2 - 4u - 16 = 4(3u - 4)(u + 1)
\]

Answer: \(4(3u - 4)(u + 1)\)

---

#### Problem 11: \( z^4 + 6z^2 - 7z^2 \)
Simplify the expression:
\[
z^4 + 6z^2 - 7z^2 = z^4 - z^2
\]
Factor out \(z^2\):
\[
z^4 - z^2 = z^2(z^2 - 1)
\]
Now, factor \(z^2 - 1\) using the difference of squares:
\[
z^2 - 1 = (z - 1)(z + 1)
\]

Thus:
\[
z^4 - z^2 = z^2(z - 1)(z + 1)
\]

Answer: \(z^2(z - 1)(z + 1)\)

---

#### Problem 12: \( 9k^2 + 66k + 21 \)
Factor out the GCF, which is \(3\):
\[
9k^2 + 66k + 21 = 3(3k^2 + 22k + 7)
\]
Now, factor \(3k^2 + 22k + 7\). Find two numbers that multiply to \(3 \cdot 7 = 21\) and add to \(22\). These numbers are \(21\) and \(1\).

\[
3k^2 + 22k + 7 = 3k^2 + 21k + k + 7
\]
\[
= 3k(k + 7) + 1(k + 7)
\]
\[
= (3k + 1)(k + 7)
\]

Thus:
\[
9k^2 + 66k + 21 = 3(3k + 1)(k + 7)
\]

Answer: \(3(3k + 1)(k + 7)\)

---

#### Problem 13: \( 6x^2 - 7xy - 3y^2 \)
This is a quadratic in terms of \(x\). Treat \(y\) as a constant. Find two numbers that multiply to \(6 \cdot (-3) = -18\) and add to \(-7\). These numbers are \(-9\) and \(2\).

\[
6x^2 - 7xy - 3y^2 = 6x^2 - 9xy + 2xy - 3y^2
\]
\[
= 3x(2x - 3y) + y(2x - 3y)
\]
\[
= (3x + y)(2x - 3y)
\]

Answer: \((3x + y)(2x - 3y)\)

---

#### Problem 14: \( 6b^2 - 15b - 9 \)
Factor out the GCF, which is \(3\):
\[
6b^2 - 15b - 9 = 3(2b^2 - 5b - 3)
\]
Now, factor \(2b^2 - 5b - 3\). Find two numbers that multiply to \(2 \cdot (-3) = -6\) and add to \(-5\). These numbers are \(-6\) and \(1\).

\[
2b^2 - 5b - 3 = 2b^2 - 6b + b - 3
\]
\[
= 2b(b - 3) + 1(b - 3)
\]
\[
= (2b + 1)(b - 3)
\]

Thus:
\[
6b^2 - 15b - 9 = 3(2b + 1)(b - 3)
\]

Answer: \(3(2b + 1)(b - 3)\)

---

#### Problem 15: \( 4w^2 - 17w - 15 \)
Find two numbers that multiply to \(4 \cdot (-15) = -60\) and add to \(-17\). These numbers are \(-20\) and \(3\).

\[
4w^2 - 17w - 15 = 4w^2 - 20w + 3w - 15
\]
\[
= 4w(w - 5) + 3(w - 5)
\]
\[
= (4w + 3)(w - 5)
\]

Answer: \((4w + 3)(w - 5)\)

---

#### Problem 16: \( 4y^2 - 17y + 15 \)
Find two numbers that multiply to \(4 \cdot 15 = 60\) and add to \(-17\). These numbers are \(-12\) and \(-5\).

\[
4y^2 - 17y + 15 = 4y^2 - 12y - 5y + 15
\]
\[
= 4y(y - 3) - 5(y - 3)
\]
\[
= (4y - 5)(y - 3)
\]

Answer: \((4y - 5)(y - 3)\)

---

#### Problem 17: \( 15m^3 + 9m^2 - 6m \)
Factor out the GCF, which is \(3m\):
\[
15m^3 + 9m^2 - 6m = 3m(5m^2 + 3m - 2)
\]
Now, factor \(5m^2 + 3m - 2\). Find two numbers that multiply to \(5 \cdot (-2) = -10\) and add to \(3\). These numbers are \(5\) and \(-2\).

\[
5m^2 + 3m - 2 = 5m^2 + 5m - 2m - 2
\]
\[
= 5m(m + 1) - 2(m + 1)
\]
\[
= (5m - 2)(m + 1)
\]

Thus:
\[
15m^3 + 9m^2 - 6m = 3m(5m - 2)(m + 1)
\]

Answer: \(3m(5m - 2)(m + 1)\)

---

#### Problem 18: \( z^4 + 2z^2 - 1 \)
Let \(u = z^2\). Then the expression becomes:
\[
u^2 + 2u - 1
\]
This does not factor nicely over the integers. Thus, the original expression remains:
\[
z^4 + 2z^2 - 1
\]

Answer: \(z^4 + 2z^2 - 1\)

---

#### Problem 19: \( h^3 - k^3 \)
Use the difference of cubes formula:
\[
h^3 - k^3 = (h - k)(h^2 + hk + k^2)
\]

Answer: \((h - k)(h^2 + hk + k^2)\)

---

#### Problem 20: \( 8p^3 + r^3 \)
Use the sum of cubes formula:
\[
8p^3 + r^3 = (2p)^3 + r^3 = (2p + r)((2p)^2 - (2p)(r) + r^2)
\]
\[
= (2p + r)(4p^2 - 2pr + r^2)
\]

Answer: \((2p + r)(4p^2 - 2pr + r^2)\)

---

#### Problem 21: \( 125h^3 - 27k^3 \)
Use the difference of cubes formula:
\[
125h^3 - 27k^3 = (5h)^3 - (3k)^3 = (5h - 3k)((5h)^2 + (5h)(3k) + (3k)^2)
\]
\[
= (5h - 3k)(25h^2 + 15hk + 9k^2)
\]

Answer: \((5h - 3k)(25h^2 + 15hk + 9k^2)\)

---

#### Problem 22: \( 27c^6 - 8d^3 \)
Rewrite as a difference of cubes:
\[
27c^6 - 8d^3 = (3c^2)^3 - (2d)^3 = (3c^2 - 2d)((3c^2)^2 + (3c^2)(2d) + (2d)^2)
\]
\[
= (3c^2 - 2d)(9c^4 + 6c^2d + 4d^2)
\]

Answer: \((3c^2 - 2d)(9c^4 + 6c^2d + 4d^2)\)

---

#### Problem 23: \( 250a^3 + 2b^3 \)
Factor out the GCF, which is \(2\):
\[
250a^3 + 2b^3 = 2(125a^3 + b^3)
\]
Now, use the sum of cubes formula on \(125a^3 + b^3\):
\[
125a^3 + b^3 = (5a)^3 + b^3 = (5a + b)((5a)^2 - (5a)(b) + b^2)
\]
\[
= (5a + b)(25a^2 - 5ab + b^2)
\]

Thus:
\[
250a^3 + 2b^3 = 2(5a + b)(25a^2 - 5ab + b^2)
\]

Answer: \(2(5a + b)(25a^2 - 5ab + b^2)\)

---

#### Problem 24: \( 36w^2 - 49z^2 \)
Use the difference of squares formula:
\[
36w^2 - 49z^2 = (6w)^2 - (7z)^2 = (6w - 7z)(6w + 7z)
\]

Answer: \((6w - 7z)(6w + 7z)\)

---

#### Problem 25: \( 5x^2 - 45 \)
Factor out the GCF, which is \(5\):
\[
5x^2 - 45 = 5(x^2 - 9)
\]
Now, use the difference of squares formula on \(x^2 - 9\):
\[
x^2 - 9 = (x - 3)(x + 3)
\]

Thus:
\[
5x^2 - 45 = 5(x - 3)(x + 3)
\]

Answer: \(5(x - 3)(x + 3)\)

---

#### Problem 26: \( a^3x - 16ax^3 \)
Factor out the GCF, which is \(ax\):
\[
a^3x - 16ax^3 = ax(a^2 - 16x^2)
\]
Now, use the difference of squares formula on \(a^2 - 16x^2\):
\[
a^2 - 16x^2 = (a - 4x)(a + 4x)
\]

Thus:
\[
a^3x - 16ax^3 = ax(a - 4x)(a + 4x)
\]

Answer: \(ax(a - 4x)(a + 4x)\)

---

#### Problem 27: \( 16a^4 - b^4 \)
Use the difference of squares formula twice:
\[
16a^4 - b^4 = (4a^2)^2 - (b^2)^2 = (4a^2 - b^2)(4a^2 + b^2)
\]
Now, factor \(4a^2 - b^2\) using the difference of squares again:
\[
4a^2 - b^2 = (2a - b)(2a + b)
\]

Thus:
\[
16a^4 - b^4 = (2a - b)(2a + b)(4a^2 + b^2)
\]

Answer: \((2a - b)(2a + b)(4a^2 + b^2)\)

---

#### Problem 28: \( 3x^7 + 36x^5 + 108x^3 \)
Factor out the GCF, which is \(3x^3\):
\[
3x^7 + 36x^5 + 108x^3 = 3x^3(x^4 + 12x^2 + 36)
\]
Now, recognize that \(x^4 + 12x^2 + 36\) is a perfect square trinomial:
\[
x^4 + 12x^2 + 36 = (x^2 + 6)^2
\]

Thus:
\[
3x^7 + 36x^5 + 108x^3 = 3x^3(x^2 + 6)^2
\]

Answer: \(3x^3(x^2 + 6)^2\)

---

#### Problem 29: \( 4u^2 + 20u + 25 \)
This is a perfect square trinomial:
\[
4u^2 + 20u + 25 = (2u)^2 + 2 \cdot 2u \cdot 5 + 5^2 = (2u + 5)^2
\]

Answer: \((2u + 5)^2\)

---

#### Problem 30: \( 121b^2 - 66bd + 9d^2 \)
This is a perfect square trinomial:
\[
121b^2 - 66bd + 9d^2 = (11b)^2 - 2 \cdot 11b \cdot 3d + (3d)^2 = (11b - 3d)^2
\]

Answer: \((11b - 3d)^2\)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ (3p - 5)(p + 1) \\
4. & \ (2y - 5)^2 \\
5. & \ 20j(20 - 6j) \\
8. & \ (4y - 1)(y - 4) \\
9. & \ -(5x - 3)(x - 3) \\
10. & \ 4(3u - 4)(u + 1) \\
11. & \ z^2(z - 1)(z + 1) \\
12. & \ 3(3k + 1)(k + 7) \\
13. & \ (3x + y)(2x - 3y) \\
14. & \ 3(2b + 1)(b - 3) \\
15. & \ (4w + 3)(w - 5) \\
16. & \ (4y - 5)(y - 3) \\
17. & \ 3m(5m - 2)(m + 1) \\
18. & \ z^4 + 2z^2 - 1 \\
19. & \ (h - k)(h^2 + hk + k^2) \\
20. & \ (2p + r)(4p^2 - 2pr + r^2) \\
21. & \ (5h - 3k)(25h^2 + 15hk + 9k^2) \\
22. & \ (3c^2 - 2d)(9c^4 + 6c^2d + 4d^2) \\
23. & \ 2(5a + b)(25a^2 - 5ab + b^2) \\
24. & \ (6w - 7z)(6w + 7z) \\
25. & \ 5(x - 3)(x + 3) \\
26. & \ ax(a - 4x)(a + 4x) \\
27. & \ (2a - b)(2a + b)(4a^2 + b^2) \\
28. & \ 3x^3(x^2 + 6)^2 \\
29. & \ (2u + 5)^2 \\
30. & \ (11b - 3d)^2 \\
\end{aligned}
}
\]