Simplifying Polynomials worksheet with 30 algebraic expressions to simplify.

Problem: Simplify each polynomial expression.

The task involves simplifying a series of polynomial expressions. We will go through each expression step by step, combining like terms and performing necessary operations such as distribution.

---

#### Expression 1: \( 21 + 3(-6x^3 - 7x^2) - 1 + x \)

1. Distribute the \( 3 \) inside the parentheses:
\[
21 + 3(-6x^3) + 3(-7x^2) - 1 + x = 21 - 18x^3 - 21x^2 - 1 + x
\]

2. Combine like terms:
- Constant terms: \( 21 - 1 = 20 \)
- \( x^3 \)-terms: \( -18x^3 \)
- \( x^2 \)-terms: \( -21x^2 \)
- \( x \)-terms: \( x \)

Result:
\[
-18x^3 - 21x^2 + x + 20
\]

Answer for Expression 1:
\[
\boxed{-18x^3 - 21x^2 + x + 20}
\]

---

#### Expression 2: \( 15x(x + 2x^2 - 7x^4) \)

1. Distribute \( 15x \) across each term inside the parentheses:
\[
15x \cdot x + 15x \cdot 2x^2 + 15x \cdot (-7x^4) = 15x^2 + 30x^3 - 105x^5
\]

2. The expression is already simplified.

Answer for Expression 2:
\[
\boxed{15x^2 + 30x^3 - 105x^5}
\]

---

#### Expression 3: \( (-x + 5x^2)x \)

1. Distribute \( x \) across each term inside the parentheses:
\[
(-x)x + (5x^2)x = -x^2 + 5x^3
\]

2. The expression is already simplified.

Answer for Expression 3:
\[
\boxed{5x^3 - x^2}
\]

---

#### Expression 4: \( 16 + 3x^3 - 7x^2 - 2 \)

1. Combine like terms:
- Constant terms: \( 16 - 2 = 14 \)
- \( x^3 \)-terms: \( 3x^3 \)
- \( x^2 \)-terms: \( -7x^2 \)

Result:
\[
3x^3 - 7x^2 + 14
\]

Answer for Expression 4:
\[
\boxed{3x^3 - 7x^2 + 14}
\]

---

#### Expression 5: \( 6 + 3(-2x^3 - 5x^2) - 2 + x \)

1. Distribute the \( 3 \) inside the parentheses:
\[
6 + 3(-2x^3) + 3(-5x^2) - 2 + x = 6 - 6x^3 - 15x^2 - 2 + x
\]

2. Combine like terms:
- Constant terms: \( 6 - 2 = 4 \)
- \( x^3 \)-terms: \( -6x^3 \)
- \( x^2 \)-terms: \( -15x^2 \)
- \( x \)-terms: \( x \)

Result:
\[
-6x^3 - 15x^2 + x + 4
\]

Answer for Expression 5:
\[
\boxed{-6x^3 - 15x^2 + x + 4}
\]

---

#### Expression 6: \( (x - 4x^2)(x + 5) \)

1. Use the distributive property (FOIL method):
\[
(x - 4x^2)(x + 5) = x(x) + x(5) - 4x^2(x) - 4x^2(5)
\]
\[
= x^2 + 5x - 4x^3 - 20x^2
\]

2. Combine like terms:
- \( x^3 \)-terms: \( -4x^3 \)
- \( x^2 \)-terms: \( x^2 - 20x^2 = -19x^2 \)
- \( x \)-terms: \( 5x \)

Result:
\[
-4x^3 - 19x^2 + 5x
\]

Answer for Expression 6:
\[
\boxed{-4x^3 - 19x^2 + 5x}
\]

---

#### Expression 7: \( 13 + 3x^3 - 7x^2 - 0 \)

1. Simplify the constant term \( -0 \):
\[
13 + 3x^3 - 7x^2
\]

2. The expression is already simplified.

Answer for Expression 7:
\[
\boxed{3x^3 - 7x^2 + 13}
\]

---

#### Expression 8: \( 15 + 2x^3 - 4x^2 - 4 \)

1. Combine like terms:
- Constant terms: \( 15 - 4 = 11 \)
- \( x^3 \)-terms: \( 2x^3 \)
- \( x^2 \)-terms: \( -4x^2 \)

Result:
\[
2x^3 - 4x^2 + 11
\]

Answer for Expression 8:
\[
\boxed{2x^3 - 4x^2 + 11}
\]

---

#### Expression 9: \( (x - 5x^2)(x + 3) \)

1. Use the distributive property (FOIL method):
\[
(x - 5x^2)(x + 3) = x(x) + x(3) - 5x^2(x) - 5x^2(3)
\]
\[
= x^2 + 3x - 5x^3 - 15x^2
\]

2. Combine like terms:
- \( x^3 \)-terms: \( -5x^3 \)
- \( x^2 \)-terms: \( x^2 - 15x^2 = -14x^2 \)
- \( x \)-terms: \( 3x \)

Result:
\[
-5x^3 - 14x^2 + 3x
\]

Answer for Expression 9:
\[
\boxed{-5x^3 - 14x^2 + 3x}
\]

---

#### Expression 10: \( 5x(x + 5x^2 - 7x^4) \)

1. Distribute \( 5x \) across each term inside the parentheses:
\[
5x \cdot x + 5x \cdot 5x^2 + 5x \cdot (-7x^4) = 5x^2 + 25x^3 - 35x^5
\]

2. The expression is already simplified.

Answer for Expression 10:
\[
\boxed{5x^2 + 25x^3 - 35x^5}
\]

---

#### Expression 11: \( (x - 11x^2)(x + 7) \)

1. Use the distributive property (FOIL method):
\[
(x - 11x^2)(x + 7) = x(x) + x(7) - 11x^2(x) - 11x^2(7)
\]
\[
= x^2 + 7x - 11x^3 - 77x^2
\]

2. Combine like terms:
- \( x^3 \)-terms: \( -11x^3 \)
- \( x^2 \)-terms: \( x^2 - 77x^2 = -76x^2 \)
- \( x \)-terms: \( 7x \)

Result:
\[
-11x^3 - 76x^2 + 7x
\]

Answer for Expression 11:
\[
\boxed{-11x^3 - 76x^2 + 7x}
\]

---

#### Expression 12: \( 7 + 4x^3 - 4x^2 - 2 \)

1. Combine like terms:
- Constant terms: \( 7 - 2 = 5 \)
- \( x^3 \)-terms: \( 4x^3 \)
- \( x^2 \)-terms: \( -4x^2 \)

Result:
\[
4x^3 - 4x^2 + 5
\]

Answer for Expression 12:
\[
\boxed{4x^3 - 4x^2 + 5}
\]

---

#### Expression 13: \( (-x + 9x^2)x \)

1. Distribute \( x \) across each term inside the parentheses:
\[
(-x)x + (9x^2)x = -x^2 + 9x^3
\]

2. The expression is already simplified.

Answer for Expression 13:
\[
\boxed{9x^3 - x^2}
\]

---

#### Expression 14: \( 11 + 3(-2x^3 - 2x^2) - 2 + x \)

1. Distribute the \( 3 \) inside the parentheses:
\[
11 + 3(-2x^3) + 3(-2x^2) - 2 + x = 11 - 6x^3 - 6x^2 - 2 + x
\]

2. Combine like terms:
- Constant terms: \( 11 - 2 = 9 \)
- \( x^3 \)-terms: \( -6x^3 \)
- \( x^2 \)-terms: \( -6x^2 \)
- \( x \)-terms: \( x \)

Result:
\[
-6x^3 - 6x^2 + x + 9
\]

Answer for Expression 14:
\[
\boxed{-6x^3 - 6x^2 + x + 9}
\]

---

#### Expression 15: \( (x - 11x^2)(x + 5) \)

1. Use the distributive property (FOIL method):
\[
(x - 11x^2)(x + 5) = x(x) + x(5) - 11x^2(x) - 11x^2(5)
\]
\[
= x^2 + 5x - 11x^3 - 55x^2
\]

2. Combine like terms:
- \( x^3 \)-terms: \( -11x^3 \)
- \( x^2 \)-terms: \( x^2 - 55x^2 = -54x^2 \)
- \( x \)-terms: \( 5x \)

Result:
\[
-11x^3 - 54x^2 + 5x
\]

Answer for Expression 15:
\[
\boxed{-11x^3 - 54x^2 + 5x}
\]

---

#### Expression 16: \( 5 + 4x^3 - 2x^2 - 3 \)

1. Combine like terms:
- Constant terms: \( 5 - 3 = 2 \)
- \( x^3 \)-terms: \( 4x^3 \)
- \( x^2 \)-terms: \( -2x^2 \)

Result:
\[
4x^3 - 2x^2 + 2
\]

Answer for Expression 16:
\[
\boxed{4x^3 - 2x^2 + 2}
\]

---

#### Expression 17: \( 16x(x + 6x^2 - 4x^4) \)

1. Distribute \( 16x \) across each term inside the parentheses:
\[
16x \cdot x + 16x \cdot 6x^2 + 16x \cdot (-4x^4) = 16x^2 + 96x^3 - 64x^5
\]

2. The expression is already simplified.

Answer for Expression 17:
\[
\boxed{16x^2 + 96x^3 - 64x^5}
\]

---

#### Expression 18: \( 20 + 5x^3 - 3x^2 - 2 \)

1. Combine like terms:
- Constant terms: \( 20 - 2 = 18 \)
- \( x^3 \)-terms: \( 5x^3 \)
- \( x^2 \)-terms: \( -3x^2 \)

Result:
\[
5x^3 - 3x^2 + 18
\]

Answer for Expression 18:
\[
\boxed{5x^3 - 3x^2 + 18}
\]

---

#### Expression 19: \( 14x(x + 2x^2 - 2x^4) \)

1. Distribute \( 14x \) across each term inside the parentheses:
\[
14x \cdot x + 14x \cdot 2x^2 + 14x \cdot (-2x^4) = 14x^2 + 28x^3 - 28x^5
\]

2. The expression is already simplified.

Answer for Expression 19:
\[
\boxed{14x^2 + 28x^3 - 28x^5}
\]

---

#### Expression 20: \( 22 + 7x^3 - 5x^2 - 3 \)

1. Combine like terms:
- Constant terms: \( 22 - 3 = 19 \)
- \( x^3 \)-terms: \( 7x^3 \)
- \( x^2 \)-terms: \( -5x^2 \)

Result:
\[
7x^3 - 5x^2 + 19
\]

Answer for Expression 20:
\[
\boxed{7x^3 - 5x^2 + 19}
\]

---

#### Expression 21: \( 18x(x + 6x^2 - 7x^4) \)

1. Distribute \( 18x \) across each term inside the parentheses:
\[
18x \cdot x + 18x \cdot 6x^2 + 18x \cdot (-7x^4) = 18x^2 + 108x^3 - 126x^5
\]

2. The expression is already simplified.

Answer for Expression 21:
\[
\boxed{18x^2 + 108x^3 - 126x^5}
\]

---

#### Expression 22: \( (-x + 4x^2)x \)

1. Distribute \( x \) across each term inside the parentheses:
\[
(-x)x + (4x^2)x = -x^2 + 4x^3
\]

2. The expression is already simplified.

Answer for Expression 22:
\[
\boxed{4x^3 - x^2}
\]

---

#### Expression 23: \( 12 + 2x^3 - 3x^2 - 4 \)

1. Combine like terms:
- Constant terms: \( 12 - 4 = 8 \)
- \( x^3 \)-terms: \( 2x^3 \)
- \( x^2 \)-terms: \( -3x^2 \)

Result:
\[
2x^3 - 3x^2 + 8
\]

Answer for Expression 23:
\[
\boxed{2x^3 - 3x^2 + 8}
\]

---

#### Expression 24: \( 8x(x + 7x^2 - 2x^4) \)

1. Distribute \( 8x \) across each term inside the parentheses:
\[
8x \cdot x + 8x \cdot 7x^2 + 8x \cdot (-2x^4) = 8x^2 + 56x^3 - 16x^5
\]

2. The expression is already simplified.

Answer for Expression 24:
\[
\boxed{8x^2 + 56x^3 - 16x^5}
\]

---

#### Expression 25: \( 12 + 2x^3 - 6x^2 - 3 \)

1. Combine like terms:
- Constant terms: \( 12 - 3 = 9 \)
- \( x^3 \)-terms: \( 2x^3 \)
- \( x^2 \)-terms: \( -6x^2 \)

Result:
\[
2x^3 - 6x^2 + 9
\]

Answer for Expression 25:
\[
\boxed{2x^3 - 6x^2 + 9}
\]

---

#### Expression 26: \( (x - 20x^2)(x + 2) \)

1. Use the distributive property (FOIL method):
\[
(x - 20x^2)(x + 2) = x(x) + x(2) - 20x^2(x) - 20x^2(2)
\]
\[
= x^2 + 2x - 20x^3 - 40x^2
\]

2. Combine like terms:
- \( x^3 \)-terms: \( -20x^3 \)
- \( x^2 \)-terms: \( x^2 - 40x^2 = -39x^2 \)
- \( x \)-terms: \( 2x \)

Result:
\[
-20x^3 - 39x^2 + 2x
\]

Answer for Expression 26:
\[
\boxed{-20x^3 - 39x^2 + 2x}
\]

---

#### Expression 27: \( (x - 13x^2)(x + 3) \)

1. Use the distributive property (FOIL method):
\[
(x - 13x^2)(x + 3) = x(x) + x(3) - 13x^2(x) - 13x^2(3)
\]
\[
= x^2 + 3x - 13x^3 - 39x^2
\]

2. Combine like terms:
- \( x^3 \)-terms: \( -13x^3 \)
- \( x^2 \)-terms: \( x^2 - 39x^2 = -38x^2 \)
- \( x \)-terms: \( 3x \)

Result:
\[
-13x^3 - 38x^2 + 3x
\]

Answer for Expression 27:
\[
\boxed{-13x^3 - 38x^2 + 3x}
\]

---

#### Expression 28: \( 22 + 3(-7x^3 - 4x^2) - 4 + x \)

1. Distribute the \( 3 \) inside the parentheses:
\[
22 + 3(-7x^3) + 3(-4x^2) - 4 + x = 22 - 21x^3 - 12x^2 - 4 + x
\]

2. Combine like terms:
- Constant terms: \( 22 - 4 = 18 \)
- \( x^3 \)-terms: \( -21x^3 \)
- \( x^2 \)-terms: \( -12x^2 \)
- \( x \)-terms: \( x \)

Result:
\[
-21x^3 - 12x^2 + x + 18
\]

Answer for Expression 28:
\[
\boxed{-21x^3 - 12x^2 + x + 18}
\]

---

#### Expression 29: \( (x - 15x^2)(x + 4) \)

1. Use the distributive property (FOIL method):
\[
(x - 15x^2)(x + 4) = x(x) + x(4) - 15x^2(x) - 15x^2(4)
\]
\[
= x^2 + 4x - 15x^3 - 60x^2
\]

2. Combine like terms:
- \( x^3 \)-terms: \( -15x^3 \)
- \( x^2 \)-terms: \( x^2 - 60x^2 = -59x^2 \)
- \( x \)-terms: \( 4x \)

Result:
\[
-15x^3 - 59x^2 + 4x
\]

Answer for Expression 29:
\[
\boxed{-15x^3 - 59x^2 + 4x}
\]

---

#### Expression 30: \( 18 + 3(-2x^3 - 4x^2) - 2 + x \)

1. Distribute the \( 3 \) inside the parentheses:
\[
18 + 3(-2x^3) + 3(-4x^2) - 2 + x = 18 - 6x^3 - 12x^2 - 2 + x
\]

2. Combine like terms:
- Constant terms: \( 18 - 2 = 16 \)
- \( x^3 \)-terms: \( -6x^3 \)
- \( x^2 \)-terms: \( -12x^2 \)
- \( x \)-terms: \( x \)

Result:
\[
-6x^3 - 12x^2 + x + 16
\]

Answer for Expression 30:
\[
\boxed{-6x^3 - 12x^2 + x + 16}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ -18x^3 - 21x^2 + x + 20 \\
2. & \ 15x^2 + 30x^3 - 105x^5 \\
3. & \ 5x^3 - x^2 \\
4. & \ 3x^3 - 7x^2 + 14 \\
5. & \ -6x^3 - 15x^2 + x + 4 \\
6. & \ -4x^3 - 19x^2 + 5x \\
7. & \ 3x^3 - 7x^2 + 13 \\
8. & \ 2x^3 - 4x^2 + 11 \\
9. & \ -5x^3 - 14x^2 + 3x \\
10. & \ 5x^2 + 25x^3 - 35x^5 \\
11. & \ -11x^3 - 76x^2 + 7x \\
12. & \ 4x^3 - 4x^2 + 5 \\
13. & \ 9x^3 - x^2 \\
14. & \ -6x^3 - 6x^2 + x + 9 \\
15. & \ -11x^3 - 54x^2 + 5x \\
16. & \ 4x^3 - 2x^2 + 2 \\
17. & \ 16x^2 + 96x^3 - 64x^5 \\
18. & \ 5x^3 - 3x^2 + 18 \\
19. & \ 14x^2 + 28x^3 - 28x^5 \\
20. & \ 7x^3 - 5x^2 + 19 \\
21. & \ 18x^2 + 108x^3 - 126x^5 \\
22. & \ 4x^3 - x^2 \\
23. & \ 2x^3 - 3x^2 + 8 \\
24. & \ 8x^2 + 56x^3 - 16x^5 \\
25. & \ 2x^3 - 6x^2 + 9 \\
26. & \ -20x^3 - 39x^2 + 2x \\
27. & \ -13x^3 - 38x^2 + 3x \\
28. & \ -21x^3 - 12x^2 + x + 18 \\
29. & \ -15x^3 - 59x^2 + 4x \\
30. & \ -6x^3 - 12x^2 + x + 16 \\
\end{aligned}
}
\]