Problem Overview:
The task involves simplifying algebraic expressions and matching the simplified results to corresponding letters in a grid. The goal is to solve each problem, simplify the expression, and then find the matching letter based on the provided answer key.
Step-by-Step Solution:
####
Simplify the Expressions:
1.
(2x² + 4x + 1) + (3x² + 6x + 5)
- Combine like terms:
\[
(2x^2 + 3x^2) + (4x + 6x) + (1 + 5) = 5x^2 + 10x + 6
\]
- Match:
U (5x² + 10x + 6)
2.
(x + 6) + (3x + 5)
- Combine like terms:
\[
(x + 3x) + (6 + 5) = 4x + 11
\]
- Match:
B (4x + 11)
3.
(x³ + 2x² − 4) + (3x² + 6)
- Combine like terms:
\[
x^3 + (2x^2 + 3x^2) + (-4 + 6) = x^3 + 5x^2 + 2
\]
- Match:
L (x³ + 5x² + 2)
4.
(4x − 2y) + (−2x + 6y)
- Combine like terms:
\[
(4x - 2x) + (-2y + 6y) = 2x + 4y
\]
- Match:
O (2x + 4y)
5.
(x² + 6x − 4) + (−x² − 6x + 4)
- Combine like terms:
\[
(x^2 - x^2) + (6x - 6x) + (-4 + 4) = 0
\]
- Match:
E (0)
6.
6x − 4 − 6x
- Combine like terms:
\[
(6x - 6x) - 4 = -4
\]
- Match:
C (-4)
7.
(2x² + 4x + 1) − (3x² + 6x + 5)
- Distribute the negative sign and combine like terms:
\[
2x^2 + 4x + 1 - 3x^2 - 6x - 5 = (2x^2 - 3x^2) + (4x - 6x) + (1 - 5) = -x^2 - 2x - 4
\]
- Match:
D (-x² − 2x − 4)
8.
(x³ + 2x² − 4) − (3x² + 6)
- Distribute the negative sign and combine like terms:
\[
x^3 + 2x^2 - 4 - 3x^2 - 6 = x^3 + (2x^2 - 3x^2) + (-4 - 6) = x^3 - x^2 - 10
\]
- Match:
N (x³ − x² − 10)
9.
x² + 6 − 6 + x²
- Combine like terms:
\[
x^2 + x^2 + (6 - 6) = 2x^2
\]
- Match:
S (2x²)
10.
(3m − 6n) + (2m + n)
- Combine like terms:
\[
(3m + 2m) + (-6n + n) = 5m - 5n
\]
- Match:
M (5m − 5n)
11.
(5x − 6) − (8x + 2)
- Distribute the negative sign and combine like terms:
\[
5x - 6 - 8x - 2 = (5x - 8x) + (-6 - 2) = -3x - 8
\]
- Match:
I (-3x − 8)
12.
(3x² + x − 4) + (−8x² − 4x + 6)
- Combine like terms:
\[
(3x^2 - 8x^2) + (x - 4x) + (-4 + 6) = -5x^2 - 3x + 2
\]
- Match:
D (-5x² − 3x + 2)
13.
(x² + 6x + 5) + (x² + 6x + 5)
- Combine like terms:
\[
(x^2 + x^2) + (6x + 6x) + (5 + 5) = 2x^2 + 12x + 10
\]
- Match:
K (2x² + 12x + 10)
14.
3x(x² + 2x − 6) + 4(x² − 6x + 2)
- Distribute and combine like terms:
\[
3x(x^2 + 2x - 6) + 4(x^2 - 6x + 2) = 3x^3 + 6x^2 - 18x + 4x^2 - 24x + 8
\]
\[
= 3x^3 + (6x^2 + 4x^2) + (-18x - 24x) + 8 = 3x^3 + 10x^2 - 42x + 8
\]
- Match:
N (3x³ + 10x² − 42x + 8)
---
Final Answer:
The solutions for each problem are matched to the corresponding letters as follows:
1.
U
2.
B
3.
L
4.
O
5.
E
6.
C
7.
D
8.
N
9.
S
10.
M
11.
I
12.
D
13.
K
14.
N
Thus, the final answer is:
\[
\boxed{U, B, L, O, E, C, D, N, S, M, I, D, K, N}
\]
Parent Tip: Review the logic above to help your child master the concept of polynomial addition and subtraction worksheet.
