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

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Let’s go through each problem step by step, identifying the correct factoring method (e.g., difference of squares, sum of cubes, trinomial factoring, grouping, etc.) and solving them.

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Factoring Polynomials Worksheet 2

We’ll solve each problem one by one.

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#### 1. $ 3p^2 - 2p - 5 $

This is a quadratic trinomial. We look for two numbers that multiply to $ 3 \times (-5) = -15 $ and add to $-2$.

Those numbers: $-5$ and $3$

Split the middle term:
$$
3p^2 - 5p + 3p - 5
= p(3p - 5) + 1(3p - 5)
= (p + 1)(3p - 5)
$$

✔ Answer: $ (p + 1)(3p - 5) $

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#### 2. $ 3x^2 - 6x - 45 $

Factor out GCF first: $ 3(x^2 - 2x - 15) $

Now factor $ x^2 - 2x - 15 $:
Find two numbers that multiply to $-15$, add to $-2$: $-5$ and $3$

So:
$$
3(x - 5)(x + 3)
$$

✔ Answer: $ 3(x - 5)(x + 3) $

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#### 3. $ 4w^2 - 15w - 25 $

Use AC method: $ 4 \times (-25) = -100 $. Find two numbers that multiply to $-100$, add to $-15$: $-20$ and $5$

Split:
$$
4w^2 - 20w + 5w - 25
= 4w(w - 5) + 5(w - 5)
= (4w + 5)(w - 5)
$$

✔ Answer: $ (4w + 5)(w - 5) $

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#### 4. $ 4y^2 - 20y + 25 $

Perfect square trinomial:

Check: $ (2y)^2 = 4y^2 $, $ 25 = 5^2 $, middle term: $ 2 \cdot 2y \cdot 5 = 20y $ → matches

So:
$$
(2y - 5)^2
$$

✔ Answer: $ (2y - 5)^2 $

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#### 5. $ 10j^3 - 130j^2 + 400j $

Factor out GCF: $ 10j(j^2 - 13j + 40) $

Now factor $ j^2 - 13j + 40 $:
Numbers that multiply to $40$, add to $-13$: $-8$ and $-5$

So:
$$
10j(j - 8)(j - 5)
$$

✔ Answer: $ 10j(j - 8)(j - 5) $

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#### 6. $ -49 + 35b - 4b^2 $

Rewriting: $ -4b^2 + 35b - 49 $

Factor out $-1$: $ - (4b^2 - 35b + 49) $

Now factor $ 4b^2 - 35b + 49 $:
AC = $ 4 \times 49 = 196 $, need two numbers that multiply to 196, add to $-35$: $-28$ and $-7$

Split:
$$
4b^2 - 28b - 7b + 49
= 4b(b - 7) -7(b - 7)
= (4b - 7)(b - 7)
$$

So original:
$$
- (4b - 7)(b - 7)
$$

✔ Answer: $ -(4b - 7)(b - 7) $

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#### 7. $ 2n^2 + 5mn + 2m^2 $

Treat as quadratic in $ n $:
Look for two numbers that multiply to $ 2 \cdot 2m^2 = 4m^2 $, add to $5m$: $4m$ and $m$

Split:
$$
2n^2 + 4mn + mn + 2m^2
= 2n(n + 2m) + m(n + 2m)
= (2n + m)(n + 2m)
$$

✔ Answer: $ (2n + m)(n + 2m) $

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#### 8. $ 4y^2 - 17y + 4 $

AC = $ 4 \times 4 = 16 $, need numbers adding to $-17$: $-16$ and $-1$

Split:
$$
4y^2 - 16y - y + 4
= 4y(y - 4) -1(y - 4)
= (4y - 1)(y - 4)
$$

✔ Answer: $ (4y - 1)(y - 4) $

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#### 9. $ -5x^2 + 18x - 9 $

Factor out $-1$: $ - (5x^2 - 18x + 9) $

Now factor $ 5x^2 - 18x + 9 $:
AC = $ 5 \times 9 = 45 $, need numbers adding to $-18$: $-15$ and $-3$

Split:
$$
5x^2 - 15x - 3x + 9
= 5x(x - 3) -3(x - 3)
= (5x - 3)(x - 3)
$$

So:
$$
- (5x - 3)(x - 3)
$$

✔ Answer: $ -(5x - 3)(x - 3) $

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#### 10. $ 12u^2 - 4u - 16 $

Factor out GCF: $ 4(3u^2 - u - 4) $

Now factor $ 3u^2 - u - 4 $:
AC = $ 3 \times -4 = -12 $, need numbers adding to $-1$: $-4$ and $3$

Split:
$$
3u^2 - 4u + 3u - 4
= u(3u - 4) + 1(3u - 4)
= (u + 1)(3u - 4)
$$

So:
$$
4(u + 1)(3u - 4)
$$

✔ Answer: $ 4(u + 1)(3u - 4) $

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#### 11. $ z^4 + 6z^2 - 7z^2 $

Wait — this seems off. Let's check: $ z^4 + 6z^2 - 7z^2 = z^4 - z^2 $

But that’s not right. Wait — likely typo? The expression is written as:

> 11. $ z^4 + 6z^2 - 7z^2 $

That simplifies to $ z^4 - z^2 $, which factors as:
$$
z^2(z^2 - 1) = z^2(z - 1)(z + 1)
$$

But perhaps it was meant to be $ z^4 + 6z^2 - 7 $? That would make more sense.

Assuming it's $ z^4 + 6z^2 - 7 $, let’s do that.

Let $ u = z^2 $, then:
$$
u^2 + 6u - 7 = (u + 7)(u - 1)
\Rightarrow (z^2 + 7)(z^2 - 1)
$$

Now $ z^2 - 1 = (z - 1)(z + 1) $

So full factorization:
$$
(z^2 + 7)(z - 1)(z + 1)
$$

✔ Answer: $ (z^2 + 7)(z - 1)(z + 1) $

*(Note: If the original was really $ z^4 + 6z^2 - 7z^2 $, then it's just $ z^4 - z^2 = z^2(z^2 - 1) = z^2(z - 1)(z + 1) $ — but likely a typo.)*

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#### 12. $ 9k^2 + 66k + 21 $

Factor out GCF: $ 3(3k^2 + 22k + 7) $

Now factor $ 3k^2 + 22k + 7 $:
AC = $ 3 \times 7 = 21 $, need numbers adding to $22$: $21$ and $1$

Split:
$$
3k^2 + 21k + k + 7
= 3k(k + 7) + 1(k + 7)
= (3k + 1)(k + 7)
$$

So:
$$
3(3k + 1)(k + 7)
$$

✔ Answer: $ 3(3k + 1)(k + 7) $

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#### 13. $ 6x^2 - 7xy - 3y^2 $

Treat as quadratic in $ x $:
AC = $ 6 \times (-3) = -18 $, need numbers adding to $-7$: $-9$ and $2$

Split:
$$
6x^2 - 9xy + 2xy - 3y^2
= 3x(2x - 3y) + y(2x - 3y)
= (3x + y)(2x - 3y)
$$

✔ Answer: $ (3x + y)(2x - 3y) $

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#### 14. $ 6b^2 - 15b - 9 $

Factor out GCF: $ 3(2b^2 - 5b - 3) $

Now factor $ 2b^2 - 5b - 3 $:
AC = $ 2 \times -3 = -6 $, need numbers adding to $-5$: $-6$ and $1$

Split:
$$
2b^2 - 6b + b - 3
= 2b(b - 3) + 1(b - 3)
= (2b + 1)(b - 3)
$$

So:
$$
3(2b + 1)(b - 3)
$$

✔ Answer: $ 3(2b + 1)(b - 3) $

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#### 15. $ 4w^2 - 17w - 15 $

AC = $ 4 \times -15 = -60 $, need numbers adding to $-17$: $-20$ and $3$

Split:
$$
4w^2 - 20w + 3w - 15
= 4w(w - 5) + 3(w - 5)
= (4w + 3)(w - 5)
$$

✔ Answer: $ (4w + 3)(w - 5) $

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#### 16. $ 4y^2 - 17y + 15 $

AC = $ 4 \times 15 = 60 $, need numbers adding to $-17$: $-12$ and $-5$

Split:
$$
4y^2 - 12y - 5y + 15
= 4y(y - 3) -5(y - 3)
= (4y - 5)(y - 3)
$$

✔ Answer: $ (4y - 5)(y - 3) $

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#### 17. $ 15m^2 + 9m - 6m $

Wait — this is $ 15m^2 + 9m - 6m = 15m^2 + 3m $

Factor: $ 3m(5m + 1) $

But maybe typo? Likely meant $ 15m^2 + 9m - 6 $?

Let’s assume it's $ 15m^2 + 9m - 6 $

Factor out GCF: $ 3(5m^2 + 3m - 2) $

Now factor $ 5m^2 + 3m - 2 $:
AC = $ 5 \times -2 = -10 $, need numbers adding to $3$: $5$ and $-2$

Split:
$$
5m^2 + 5m - 2m - 2
= 5m(m + 1) -2(m + 1)
= (5m - 2)(m + 1)
$$

So:
$$
3(5m - 2)(m + 1)
$$

✔ Answer: $ 3(5m - 2)(m + 1) $

*(If original was $ 15m^2 + 9m - 6m $, then it's $ 15m^2 + 3m = 3m(5m + 1) $ — but likely typo.)*

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#### 18. $ z^4 + 2z^2 - 1 $

Let $ u = z^2 $: $ u^2 + 2u - 1 $

This doesn't factor nicely over integers. Discriminant: $ 4 + 4 = 8 $, irrational roots.

So does not factor over integers.

✔ Answer: Prime (cannot be factored over integers)

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#### 19. $ h^3 - k^3 $

Difference of cubes!

$$
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
$$

So:
$$
(h - k)(h^2 + hk + k^2)
$$

✔ Answer: $ (h - k)(h^2 + hk + k^2) $

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#### 20. $ 8p^3 + r^3 $

Sum of cubes: $ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $

Here $ 8p^3 = (2p)^3 $, so:
$$
(2p + r)((2p)^2 - (2p)(r) + r^2)
= (2p + r)(4p^2 - 2pr + r^2)
$$

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

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#### 21. $ 125h^3 - 27k^3 $

Both are cubes: $ 125 = 5^3 $, $ 27 = 3^3 $

So:
$$
(5h)^3 - (3k)^3 = (5h - 3k)((5h)^2 + (5h)(3k) + (3k)^2)
= (5h - 3k)(25h^2 + 15hk + 9k^2)
$$

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

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#### 22. $ 27c^6 - 8d^3 $

Note: $ 27c^6 = (3c^2)^3 $, $ 8d^3 = (2d)^3 $

So:
$$
(3c^2)^3 - (2d)^3 = (3c^2 - 2d)((3c^2)^2 + (3c^2)(2d) + (2d)^2)
= (3c^2 - 2d)(9c^4 + 6c^2d + 4d^2)
$$

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

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#### 23. $ 250a^3 + 2b^3 $

Factor out GCF: $ 2(125a^3 + b^3) $

Now $ 125a^3 = (5a)^3 $, so:
$$
2[(5a)^3 + b^3] = 2(5a + b)((5a)^2 - (5a)(b) + b^2)
= 2(5a + b)(25a^2 - 5ab + b^2)
$$

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

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#### 24. $ 36w^2 - 49z^2 $

Difference of squares:
$$
(6w)^2 - (7z)^2 = (6w - 7z)(6w + 7z)
$$

✔ Answer: $ (6w - 7z)(6w + 7z) $

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#### 25. $ 5x^2 - 45 $

Factor out GCF: $ 5(x^2 - 9) = 5(x - 3)(x + 3) $

✔ Answer: $ 5(x - 3)(x + 3) $

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#### 26. $ a^3x - 16ax^3 $

Factor out GCF: $ ax(a^2 - 16x^2) $

Now $ a^2 - 16x^2 = (a - 4x)(a + 4x) $

So:
$$
ax(a - 4x)(a + 4x)
$$

✔ Answer: $ ax(a - 4x)(a + 4x) $

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#### 27. $ 16a^4 - b^4 $

Difference of squares: $ (4a^2)^2 - (b^2)^2 = (4a^2 - b^2)(4a^2 + b^2) $

Now $ 4a^2 - b^2 = (2a - b)(2a + b) $

So:
$$
(2a - b)(2a + b)(4a^2 + b^2)
$$

✔ Answer: $ (2a - b)(2a + b)(4a^2 + b^2) $

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#### 28. $ 3x^7 + 36x^5 + 108x^3 $

Factor out GCF: $ 3x^3(x^4 + 12x^2 + 36) $

Now $ x^4 + 12x^2 + 36 = (x^2)^2 + 2 \cdot 6 \cdot x^2 + 6^2 = (x^2 + 6)^2 $

So:
$$
3x^3(x^2 + 6)^2
$$

✔ Answer: $ 3x^3(x^2 + 6)^2 $

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#### 29. $ 4u^2 + 20u + 25 $

Perfect square trinomial:
$$
(2u)^2 + 2 \cdot 2u \cdot 5 + 5^2 = (2u + 5)^2
$$

✔ Answer: $ (2u + 5)^2 $

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#### 30. $ 121b^2 - 66bd + 9d^2 $

Check: $ 121b^2 = (11b)^2 $, $ 9d^2 = (3d)^2 $, middle term: $ 2 \cdot 11b \cdot 3d = 66bd $ → yes!

So:
$$
(11b - 3d)^2
$$

✔ Answer: $ (11b - 3d)^2 $

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✔ Final Answers Summary:

| Problem | Factored Form |
|--------|----------------|
| 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 | $ -(4b - 7)(b - 7) $ |
| 7 | $ (2n + m)(n + 2m) $ |
| 8 | $ (4y - 1)(y - 4) $ |
| 9 | $ -(5x - 3)(x - 3) $ |
| 10 | $ 4(u + 1)(3u - 4) $ |
| 11 | $ (z^2 + 7)(z - 1)(z + 1) $ *(assuming typo)* |
| 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 | $ 3(5m - 2)(m + 1) $ *(assuming typo)* |
| 18 | Prime (no integer factors) |
| 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 $ |

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