Problem Analysis and Solution
The task involves performing polynomial operations: addition, subtraction, and simplification. Let's solve each problem step by step.
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####
Problem 1: Find the sum.
\[
(2x^2 + 5x - 7) + (3 - 4x^2 + 6x)
\]
Step-by-Step Solution:
1. Combine like terms:
- For \(x^2\) terms: \(2x^2 - 4x^2 = -2x^2\)
- For \(x\) terms: \(5x + 6x = 11x\)
- For constant terms: \(-7 + 3 = -4\)
2. Write the simplified expression:
\[
-2x^2 + 11x - 4
\]
Correct Answer:
\[
\boxed{A}
\]
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####
Problem 2: Find the sum.
\[
(3y^2 + y^3 - 5) + (4y^2 - 4y + 2y^3 + 8)
\]
Step-by-Step Solution:
1. Combine like terms:
- For \(y^3\) terms: \(y^3 + 2y^3 = 3y^3\)
- For \(y^2\) terms: \(3y^2 + 4y^2 = 7y^2\)
- For \(y\) terms: \(-4y\) (no other \(y\) terms)
- For constant terms: \(-5 + 8 = 3\)
2. Write the simplified expression:
\[
3y^3 + 7y^2 - 4y + 3
\]
Correct Answer:
\[
\boxed{B}
\]
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####
Problem 3: Find the difference.
\[
(3 - 2x + 2x^2) - (4x - 5 + 3x^2)
\]
Step-by-Step Solution:
1. Distribute the negative sign across the second polynomial:
\[
(3 - 2x + 2x^2) - 4x + 5 - 3x^2
\]
2. Combine like terms:
- For \(x^2\) terms: \(2x^2 - 3x^2 = -x^2\)
- For \(x\) terms: \(-2x - 4x = -6x\)
- For constant terms: \(3 + 5 = 8\)
3. Write the simplified expression:
\[
-x^2 - 6x + 8
\]
Correct Answer:
\[
\boxed{A}
\]
---
####
Problem 4: Simplify the expression.
\[
(4n^4 - 8n + 4) - (8n^2 + 4n^4 + 1)
\]
Step-by-Step Solution:
1. Distribute the negative sign across the second polynomial:
\[
4n^4 - 8n + 4 - 8n^2 - 4n^4 - 1
\]
2. Combine like terms:
- For \(n^4\) terms: \(4n^4 - 4n^4 = 0\)
- For \(n^2\) terms: \(-8n^2\) (no other \(n^2\) terms)
- For \(n\) terms: \(-8n\) (no other \(n\) terms)
- For constant terms: \(4 - 1 = 3\)
3. Write the simplified expression:
\[
-8n^2 - 8n + 3
\]
Correct Answer:
\[
\boxed{D}
\]
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####
Problem 5: Simplify each expression.
\[
(6b^3 + 6 - b^4) - (8b^3 - 6b^4 + 2)
\]
Step-by-Step Solution:
1. Distribute the negative sign across the second polynomial:
\[
6b^3 + 6 - b^4 - 8b^3 + 6b^4 - 2
\]
2. Combine like terms:
- For \(b^4\) terms: \(-b^4 + 6b^4 = 5b^4\)
- For \(b^3\) terms: \(6b^3 - 8b^3 = -2b^3\)
- For constant terms: \(6 - 2 = 4\)
3. Write the simplified expression:
\[
5b^4 - 2b^3 + 4
\]
Correct Answer:
\[
\boxed{A}
\]
---
####
Problem 6: Simplify each expression.
\[
(6x + 4x^4 - 3x^2) + (7x^4 + 5x^2 + 8x)
\]
Step-by-Step Solution:
1. Combine like terms:
- For \(x^4\) terms: \(4x^4 + 7x^4 = 11x^4\)
- For \(x^2\) terms: \(-3x^2 + 5x^2 = 2x^2\)
- For \(x\) terms: \(6x + 8x = 14x\)
2. Write the simplified expression:
\[
11x^4 + 2x^2 + 14x
\]
Final Answer for Problem 6:
\[
\boxed{11x^4 + 2x^2 + 14x}
\]
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Summary of Answers:
1. \(\boxed{A}\)
2. \(\boxed{B}\)
3. \(\boxed{A}\)
4. \(\boxed{D}\)
5. \(\boxed{A}\)
6. \(\boxed{11x^4 + 2x^2 + 14x}\)
Parent Tip: Review the logic above to help your child master the concept of multiplying dividing polynomials worksheet.
