To simplify the given polynomial expressions, we need to combine like terms and perform any necessary operations such as distributing or factoring. Let's solve each problem step by step.
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1) \( 4x^2 + 2 - 5x^2 + 6x^8 - 13 - x^8 \)
- Combine like terms:
- \( x^8 \)-terms: \( 6x^8 - x^8 = 5x^8 \)
- \( x^2 \)-terms: \( 4x^2 - 5x^2 = -x^2 \)
- Constant terms: \( 2 - 13 = -11 \)
- Simplified expression:
\[
5x^8 - x^2 - 11
\]
---
2) \( 2y^2 + 3y - 5y^2 + y^2 \)
- Combine like terms:
- \( y^2 \)-terms: \( 2y^2 - 5y^2 + y^2 = -2y^2 \)
- \( y \)-terms: \( 3y \) (no other \( y \)-terms to combine)
- Simplified expression:
\[
-2y^2 + 3y
\]
---
3) \( -(9t^2 - 2) + 15 - 8t^2 - 2 \)
- Distribute the negative sign:
\[
-9t^2 + 2 + 15 - 8t^2 - 2
\]
- Combine like terms:
- \( t^2 \)-terms: \( -9t^2 - 8t^2 = -17t^2 \)
- Constant terms: \( 2 + 15 - 2 = 15 \)
- Simplified expression:
\[
-17t^2 + 15
\]
---
4) \( u^4 - u^2(3u^2 - 2) \)
- Distribute \( u^2 \) in the second term:
\[
u^4 - (3u^4 - 2u^2)
\]
\[
u^4 - 3u^4 + 2u^2
\]
- Combine like terms:
- \( u^4 \)-terms: \( u^4 - 3u^4 = -2u^4 \)
- \( u^2 \)-terms: \( 2u^2 \)
- Simplified expression:
\[
-2u^4 + 2u^2
\]
---
5) \( 14 - 5(2p^2 - 12p^3) - 3p^3 + 8 \)
- Distribute the \( -5 \):
\[
14 - 10p^2 + 60p^3 - 3p^3 + 8
\]
- Combine like terms:
- \( p^3 \)-terms: \( 60p^3 - 3p^3 = 57p^3 \)
- \( p^2 \)-terms: \( -10p^2 \) (no other \( p^2 \)-terms to combine)
- Constant terms: \( 14 + 8 = 22 \)
- Simplified expression:
\[
57p^3 - 10p^2 + 22
\]
---
6) \( 7a^2 + 13 - 8a^2 \)
- Combine like terms:
- \( a^2 \)-terms: \( 7a^2 - 8a^2 = -a^2 \)
- Constant terms: \( 13 \)
- Simplified expression:
\[
-a^2 + 13
\]
---
7) \( -18 + 9b^2 + 6 \)
- Combine constant terms:
\[
-18 + 6 = -12
\]
- Simplified expression:
\[
9b^2 - 12
\]
---
8) \( -11 - 9(2x^2 + 7x^3) + 8x^2 + 1 \)
- Distribute the \( -9 \):
\[
-11 - 18x^2 - 63x^3 + 8x^2 + 1
\]
- Combine like terms:
- \( x^3 \)-terms: \( -63x^3 \) (no other \( x^3 \)-terms to combine)
- \( x^2 \)-terms: \( -18x^2 + 8x^2 = -10x^2 \)
- Constant terms: \( -11 + 1 = -10 \)
- Simplified expression:
\[
-63x^3 - 10x^2 - 10
\]
---
Final Answers:
\[
\boxed{
\begin{aligned}
1) & \quad 5x^8 - x^2 - 11 \\
2) & \quad -2y^2 + 3y \\
3) & \quad -17t^2 + 15 \\
4) & \quad -2u^4 + 2u^2 \\
5) & \quad 57p^3 - 10p^2 + 22 \\
6) & \quad -a^2 + 13 \\
7) & \quad 9b^2 - 12 \\
8) & \quad -63x^3 - 10x^2 - 10
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of simplify polynomials worksheets.
