Simplify polynomial expressions worksheet with 14 problems for algebra practice.

Here is the step-by-step solution for each polynomial expression. The goal is to simplify by combining like terms and distributing where necessary.

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

First, distribute the \(-x\):

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

Now combine all terms:

\[
3x^2 - 5x^3 - 4x^4 - x^2
\]

Combine like terms:

- \( -4x^4 \)
- \( -5x^3 \)
- \( (3x^2 - x^2) = 2x^2 \)

✔ Answer: \( -4x^4 - 5x^3 + 2x^2 \)

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2) \( 17k + 8k^3 - 8 - 2(k^4 - 3) \)

Distribute the \(-2\):

\[
-2(k^4 - 3) = -2k^4 + 6
\]

Now combine all terms:

\[
17k + 8k^3 - 8 - 2k^4 + 6
\]

Combine constants: \(-8 + 6 = -2\)

✔ Answer: \( -2k^4 + 8k^3 + 17k - 2 \)

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3) \( 17x^3 - 7x^4 - 18x^3 + 10x^4 \)

Group like terms:

- \( (-7x^4 + 10x^4) = 3x^4 \)
- \( (17x^3 - 18x^3) = -x^3 \)

✔ Answer: \( 3x^4 - x^3 \)

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4) \( 10 + 2y^3 - (8y^3 - y^2 + 5) \)

Distribute the minus sign:

\[
- (8y^3 - y^2 + 5) = -8y^3 + y^2 - 5
\]

Now combine:

\[
10 + 2y^3 - 8y^3 + y^2 - 5
\]

Combine like terms:

- Constants: \(10 - 5 = 5\)
- \( y^2 \): only one term → \( y^2 \)
- \( y^3 \): \(2y^3 - 8y^3 = -6y^3\)

✔ Answer: \( -6y^3 + y^2 + 5 \)

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5) \( (5z^3 + 8z - 8z^2) - (8z^3 - 8z^2 + 10z) \)

Distribute the minus sign:

\[
= 5z^3 + 8z - 8z^2 - 8z^3 + 8z^2 - 10z
\]

Combine like terms:

- \( z^3 \): \(5z^3 - 8z^3 = -3z^3\)
- \( z^2 \): \(-8z^2 + 8z^2 = 0\)
- \( z \): \(8z - 10z = -2z\)

✔ Answer: \( -3z^3 - 2z \)

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6) \( 7 - 7x^3 + 6x - 10x^2 + 15x^3 \)

Group like terms:

- \( x^3 \): \(-7x^3 + 15x^3 = 8x^3\)
- \( x^2 \): \(-10x^2\)
- \( x \): \(6x\)
- Constant: \(7\)

✔ Answer: \( 8x^3 - 10x^2 + 6x + 7 \)

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7) \( 8m^4 + m - 2 - 10 - 5m^2 + 3m^4 \)

Group like terms:

- \( m^4 \): \(8m^4 + 3m^4 = 11m^4\)
- \( m^2 \): \(-5m^2\)
- \( m \): \(m\)
- Constants: \(-2 - 10 = -12\)

✔ Answer: \( 11m^4 - 5m^2 + m - 12 \)

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8) \( 5x^2 + 5 - 2x^2 + 3x^3 - 10 \)

Group like terms:

- \( x^3 \): \(3x^3\)
- \( x^2 \): \(5x^2 - 2x^2 = 3x^2\)
- Constants: \(5 - 10 = -5\)

✔ Answer: \( 3x^3 + 3x^2 - 5 \)

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9) \( (6x^3 - 4) + 3(x^2 - 4x^3) \)

Distribute the 3:

\[
3(x^2 - 4x^3) = 3x^2 - 12x^3
\]

Now combine:

\[
6x^3 - 4 + 3x^2 - 12x^3
\]

Group:

- \( x^3 \): \(6x^3 - 12x^3 = -6x^3\)
- \( x^2 \): \(3x^2\)
- Constant: \(-4\)

✔ Answer: \( -6x^3 + 3x^2 - 4 \)

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10) \( (5x^2 - 2x^3) - 2(x^2 + x^3) \)

Distribute the \(-2\):

\[
-2(x^2 + x^3) = -2x^2 - 2x^3
\]

Now combine:

\[
5x^2 - 2x^3 - 2x^2 - 2x^3
\]

Group:

- \( x^3 \): \(-2x^3 - 2x^3 = -4x^3\)
- \( x^2 \): \(5x^2 - 2x^2 = 3x^2\)

✔ Answer: \( -4x^3 + 3x^2 \)

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11) \( 2x^2 - 4x^4 + 10x^4 - 11x^3 \)

Group like terms:

- \( x^4 \): \(-4x^4 + 10x^4 = 6x^4\)
- \( x^3 \): \(-11x^3\)
- \( x^2 \): \(2x^2\)

✔ Answer: \( 6x^4 - 11x^3 + 2x^2 \)

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12) \( 5x^3 - (4x^4 + 2x) - 4x^3 \)

Distribute the minus sign:

\[
- (4x^4 + 2x) = -4x^4 - 2x
\]

Now combine:

\[
5x^3 - 4x^4 - 2x - 4x^3
\]

Group:

- \( x^4 \): \(-4x^4\)
- \( x^3 \): \(5x^3 - 4x^3 = x^3\)
- \( x \): \(-2x\)

✔ Answer: \( -4x^4 + x^3 - 2x \)

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13) \( 17x - 3x^2 - 2(x - 2x^2) \)

Distribute the \(-2\):

\[
-2(x - 2x^2) = -2x + 4x^2
\]

Now combine:

\[
17x - 3x^2 - 2x + 4x^2
\]

Group:

- \( x^2 \): \(-3x^2 + 4x^2 = x^2\)
- \( x \): \(17x - 2x = 15x\)

✔ Answer: \( x^2 + 15x \)

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14) \( 5x^4 - 25 + 5(6x^4 + 5) \)

Distribute the 5:

\[
5(6x^4 + 5) = 30x^4 + 25
\]

Now combine:

\[
5x^4 - 25 + 30x^4 + 25
\]

Group:

- \( x^4 \): \(5x^4 + 30x^4 = 35x^4\)
- Constants: \(-25 + 25 = 0\)

✔ Answer: \( 35x^4 \)

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## ✔ Final Answers Summary:

1) \( -4x^4 - 5x^3 + 2x^2 \)
2) \( -2k^4 + 8k^3 + 17k - 2 \)
3) \( 3x^4 - x^3 \)
4) \( -6y^3 + y^2 + 5 \)
5) \( -3z^3 - 2z \)
6) \( 8x^3 - 10x^2 + 6x + 7 \)
7) \( 11m^4 - 5m^2 + m - 12 \)
8) \( 3x^3 + 3x^2 - 5 \)
9) \( -6x^3 + 3x^2 - 4 \)
10) \( -4x^3 + 3x^2 \)
11) \( 6x^4 - 11x^3 + 2x^2 \)
12) \( -4x^4 + x^3 - 2x \)
13) \( x^2 + 15x \)
14) \( 35x^4 \)

Let me know if you’d like these written in standard form (descending powers) or with explanations for specific steps!