Simplifying Polynomials worksheet with 12 algebraic expressions for practice.

To solve the problem of simplifying each polynomial expression, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. Let's go through each expression step by step.

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Problem 1: \( 10x^3 - 2x^2 + 5x^3 - 6x^2 \)

- Combine the \( x^3 \) terms: \( 10x^3 + 5x^3 = 15x^3 \)
- Combine the \( x^2 \) terms: \( -2x^2 - 6x^2 = -8x^2 \)

So, the simplified expression is:
\[
\boxed{15x^3 - 8x^2}
\]

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Problem 2: \( 5t^5 - 2t^2 + 5t^5 - 6t^2 \)

- Combine the \( t^5 \) terms: \( 5t^5 + 5t^5 = 10t^5 \)
- Combine the \( t^2 \) terms: \( -2t^2 - 6t^2 = -8t^2 \)

So, the simplified expression is:
\[
\boxed{10t^5 - 8t^2}
\]

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Problem 3: \( 2x^3 - 7x^2 + 2x^3 - 6x^2 \)

- Combine the \( x^3 \) terms: \( 2x^3 + 2x^3 = 4x^3 \)
- Combine the \( x^2 \) terms: \( -7x^2 - 6x^2 = -13x^2 \)

So, the simplified expression is:
\[
\boxed{4x^3 - 13x^2}
\]

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Problem 4: \( 10n^4 - 3n^2 + 9n^2 - 5n^2 \)

- Combine the \( n^4 \) terms: \( 10n^4 \) (no other \( n^4 \) terms)
- Combine the \( n^2 \) terms: \( -3n^2 + 9n^2 - 5n^2 = 1n^2 \)

So, the simplified expression is:
\[
\boxed{10n^4 - n^2}
\]

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Problem 5: \( 10x^2 + 4x^2 + 6x^2 - 3x^2 \)

- Combine the \( x^2 \) terms: \( 10x^2 + 4x^2 + 6x^2 - 3x^2 = 17x^2 \)

So, the simplified expression is:
\[
\boxed{17x^2}
\]

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

- Combine the \( x^5 \) terms: \( -2x^5 \) (no other \( x^5 \) terms)
- Combine the \( x^3 \) terms: \( 3x^3 + 5x^3 = 8x^3 \)
- Combine the \( x^2 \) terms: \( 2x^2 \) (no other \( x^2 \) terms)

So, the simplified expression is:
\[
\boxed{-2x^5 + 8x^3 + 2x^2}
\]

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Problem 7: \( 3(9x^3 - 4x^2 + x^3 - 2x^2) \)

First, simplify inside the parentheses:
- Combine the \( x^3 \) terms: \( 9x^3 + x^3 = 10x^3 \)
- Combine the \( x^2 \) terms: \( -4x^2 - 2x^2 = -6x^2 \)

So, the expression inside the parentheses becomes:
\[
10x^3 - 6x^2
\]

Now distribute the 3:
\[
3(10x^3 - 6x^2) = 3 \cdot 10x^3 - 3 \cdot 6x^2 = 30x^3 - 18x^2
\]

So, the simplified expression is:
\[
\boxed{30x^3 - 18x^2}
\]

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Problem 8: \( 4(2x^3 - 6x^2) + 2x^3 - 7x^2 \)

First, distribute the 4:
\[
4(2x^3 - 6x^2) = 4 \cdot 2x^3 - 4 \cdot 6x^2 = 8x^3 - 24x^2
\]

Now add the remaining terms:
\[
8x^3 - 24x^2 + 2x^3 - 7x^2
\]

Combine the \( x^3 \) terms: \( 8x^3 + 2x^3 = 10x^3 \)
Combine the \( x^2 \) terms: \( -24x^2 - 7x^2 = -31x^2 \)

So, the simplified expression is:
\[
\boxed{10x^3 - 31x^2}
\]

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Problem 9: \( 10t^3 - 6t^2 + 3t^3 - 9t^2 \)

- Combine the \( t^3 \) terms: \( 10t^3 + 3t^3 = 13t^3 \)
- Combine the \( t^2 \) terms: \( -6t^2 - 9t^2 = -15t^2 \)

So, the simplified expression is:
\[
\boxed{13t^3 - 15t^2}
\]

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

- Combine the \( x^3 \) terms: \( 5x^3 \) (no other \( x^3 \) terms)
- Combine the \( x^2 \) terms: \( x^2 - 5x^2 - 8x^2 = -12x^2 \)

So, the simplified expression is:
\[
\boxed{5x^3 - 12x^2}
\]

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Problem 11: \( 7x^2 - 6x^3 + 5x^3 - 4x^2 \)

- Combine the \( x^3 \) terms: \( -6x^3 + 5x^3 = -x^3 \)
- Combine the \( x^2 \) terms: \( 7x^2 - 4x^2 = 3x^2 \)

So, the simplified expression is:
\[
\boxed{-x^3 + 3x^2}
\]

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Problem 12: \( 11x^3 + 6x^2 + 3x^3 - x^2 \)

- Combine the \( x^3 \) terms: \( 11x^3 + 3x^3 = 14x^3 \)
- Combine the \( x^2 \) terms: \( 6x^2 - x^2 = 5x^2 \)

So, the simplified expression is:
\[
\boxed{14x^3 + 5x^2}
\]

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Final Answers:

1. \( \boxed{15x^3 - 8x^2} \)
2. \( \boxed{10t^5 - 8t^2} \)
3. \( \boxed{4x^3 - 13x^2} \)
4. \( \boxed{10n^4 - n^2} \)
5. \( \boxed{17x^2} \)
6. \( \boxed{-2x^5 + 8x^3 + 2x^2} \)
7. \( \boxed{30x^3 - 18x^2} \)
8. \( \boxed{10x^3 - 31x^2} \)
9. \( \boxed{13t^3 - 15t^2} \)
10. \( \boxed{5x^3 - 12x^2} \)
11. \( \boxed{-x^3 + 3x^2} \)
12. \( \boxed{14x^3 + 5x^2} \)

Boxed Final Answer:
\[
\boxed{
\begin{aligned}
1. & \ 15x^3 - 8x^2 \\
2. & \ 10t^5 - 8t^2 \\
3. & \ 4x^3 - 13x^2 \\
4. & \ 10n^4 - n^2 \\
5. & \ 17x^2 \\
6. & \ -2x^5 + 8x^3 + 2x^2 \\
7. & \ 30x^3 - 18x^2 \\
8. & \ 10x^3 - 31x^2 \\
9. & \ 13t^3 - 15t^2 \\
10. & \ 5x^3 - 12x^2 \\
11. & \ -x^3 + 3x^2 \\
12. & \ 14x^3 + 5x^2 \\
\end{aligned}
}
\]