Simplifying Polynomials worksheet with 20 algebra problems for practice.

To solve the given problems, we will simplify each polynomial expression step by step. Let's go through each problem:

---

Problem 1: \( 3x^2 - 5x^3 - x(2x^2 + 4x) \)

1. Distribute \( x \) in the term \( x(2x^2 + 4x) \):
\[
x(2x^2 + 4x) = 2x^3 + 4x^2
\]

2. Substitute back into the expression:
\[
3x^2 - 5x^3 - (2x^3 + 4x^2)
\]

3. Distribute the negative sign:
\[
3x^2 - 5x^3 - 2x^3 - 4x^2
\]

4. Combine like terms:
\[
(-5x^3 - 2x^3) + (3x^2 - 4x^2) = -7x^3 - x^2
\]

Answer:
\[
\boxed{-7x^3 - x^2}
\]

---

Problem 2: \( 6(8r - 5) \)

1. Distribute the 6:
\[
6 \cdot 8r - 6 \cdot 5 = 48r - 30
\]

Answer:
\[
\boxed{48r - 30}
\]

---

Problem 3: \( (2v + 6)(4v) \)

1. Distribute \( 4v \) to both terms in the parentheses:
\[
4v \cdot 2v + 4v \cdot 6 = 8v^2 + 24v
\]

Answer:
\[
\boxed{8v^2 + 24v}
\]

---

Problem 4: \( a^2 - 2a + 5a^3 + 1 - 10a \)

1. Rearrange the terms to group like terms:
\[
5a^3 + a^2 + (-2a - 10a) + 1 = 5a^3 + a^2 - 12a + 1
\]

Answer:
\[
\boxed{5a^3 + a^2 - 12a + 1}
\]

---

Problem 5: \( x(-2x + 5x^2) \)

1. Distribute \( x \):
\[
x \cdot (-2x) + x \cdot 5x^2 = -2x^2 + 5x^3
\]

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

---

Problem 6: \( 2x^3 - 4x^3 + 2x^4 + 1 \)

1. Combine like terms:
\[
(2x^3 - 4x^3) + 2x^4 + 1 = -2x^3 + 2x^4 + 1
\]

Answer:
\[
\boxed{2x^4 - 2x^3 + 1}
\]

---

Problem 7: \( 3y^2(1 - y - 2y^2) \)

1. Distribute \( 3y^2 \):
\[
3y^2 \cdot 1 + 3y^2 \cdot (-y) + 3y^2 \cdot (-2y^2) = 3y^2 - 3y^3 - 6y^4
\]

Answer:
\[
\boxed{-6y^4 - 3y^3 + 3y^2}
\]

---

Problem 8: \( 10x^2 - 4 - 2x^2 + x^2 \)

1. Combine like terms:
\[
(10x^2 - 2x^2 + x^2) - 4 = 9x^2 - 4
\]

Answer:
\[
\boxed{9x^2 - 4}
\]

---

Problem 9: \( -5g(3 - 2g^2) \)

1. Distribute \( -5g \):
\[
-5g \cdot 3 + (-5g) \cdot (-2g^2) = -15g + 10g^3
\]

Answer:
\[
\boxed{10g^3 - 15g}
\]

---

Problem 10: \( 1 + 3x + 4x^2 - 5x + 8 \)

1. Combine like terms:
\[
4x^2 + (3x - 5x) + (1 + 8) = 4x^2 - 2x + 9
\]

Answer:
\[
\boxed{4x^2 - 2x + 9}
\]

---

Problem 11: \( 3a + b - 4b + 5a \)

1. Combine like terms:
\[
(3a + 5a) + (b - 4b) = 8a - 3b
\]

Answer:
\[
\boxed{8a - 3b}
\]

---

Problem 12: \( (7n + 8)(-4) \)

1. Distribute \( -4 \):
\[
-4 \cdot 7n + (-4) \cdot 8 = -28n - 32
\]

Answer:
\[
\boxed{-28n - 32}
\]

---

Problem 13: \( -2x^3 + 8x^3 + 9x^3 + 5 \)

1. Combine like terms:
\[
(-2x^3 + 8x^3 + 9x^3) + 5 = 15x^3 + 5
\]

Answer:
\[
\boxed{15x^3 + 5}
\]

---

Problem 14: \( (3x + 4)(3x) \)

1. Distribute \( 3x \):
\[
3x \cdot 3x + 3x \cdot 4 = 9x^2 + 12x
\]

Answer:
\[
\boxed{9x^2 + 12x}
\]

---

Problem 15: \( 5m^2 - 7m^3 + 3m^2 + 3 \)

1. Combine like terms:
\[
(-7m^3) + (5m^2 + 3m^2) + 3 = -7m^3 + 8m^2 + 3
\]

Answer:
\[
\boxed{-7m^3 + 8m^2 + 3}
\]

---

Problem 16: \( 3x(1 + 6x - 5) \)

1. Simplify inside the parentheses:
\[
1 + 6x - 5 = 6x - 4
\]

2. Distribute \( 3x \):
\[
3x \cdot 6x + 3x \cdot (-4) = 18x^2 - 12x
\]

Answer:
\[
\boxed{18x^2 - 12x}
\]

---

Problem 17: \( (2x - 6)(7x) \)

1. Distribute \( 7x \):
\[
7x \cdot 2x + 7x \cdot (-6) = 14x^2 - 42x
\]

Answer:
\[
\boxed{14x^2 - 42x}
\]

---

Problem 18: \( 12x + 10 - 2x - 8 \)

1. Combine like terms:
\[
(12x - 2x) + (10 - 8) = 10x + 2
\]

Answer:
\[
\boxed{10x + 2}
\]

---

Problem 19: \( 5x + 10 - 2x + 5 \)

1. Combine like terms:
\[
(5x - 2x) + (10 + 5) = 3x + 15
\]

Answer:
\[
\boxed{3x + 15}
\]

---

Problem 20: \( 2x(3x + 5) \)

1. Distribute \( 2x \):
\[
2x \cdot 3x + 2x \cdot 5 = 6x^2 + 10x
\]

Answer:
\[
\boxed{6x^2 + 10x}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ -7x^3 - x^2 \\
2. & \ 48r - 30 \\
3. & \ 8v^2 + 24v \\
4. & \ 5a^3 + a^2 - 12a + 1 \\
5. & \ 5x^3 - 2x^2 \\
6. & \ 2x^4 - 2x^3 + 1 \\
7. & \ -6y^4 - 3y^3 + 3y^2 \\
8. & \ 9x^2 - 4 \\
9. & \ 10g^3 - 15g \\
10. & \ 4x^2 - 2x + 9 \\
11. & \ 8a - 3b \\
12. & \ -28n - 32 \\
13. & \ 15x^3 + 5 \\
14. & \ 9x^2 + 12x \\
15. & \ -7m^3 + 8m^2 + 3 \\
16. & \ 18x^2 - 12x \\
17. & \ 14x^2 - 42x \\
18. & \ 10x + 2 \\
19. & \ 3x + 15 \\
20. & \ 6x^2 + 10x \\
\end{aligned}
}
\]