Wells Bates
02/24/2023 · Senior High School
\begin{tabular}{|l|l|}\hline 25. & \( (x+2 y)^{3}= \) \\ \hline 26. & \( (2 a+b)^{3}= \) \\ \hline 27. & \( (2 x-5)^{3}= \) \\ \hline 28. & \( (a-b)^{3}=a^{2}-3 a^{2} b-3 a b-b^{3} \) \\ \hline 29. & \( (3 a-5)^{3}= \) \\ \hline\end{tabular}
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To expand the expressions given in the table, we can use the binomial theorem, which states that:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
For \( n = 3 \), the expansion becomes:
\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]
Now, let's apply this to each expression:
### 25. \( (x + 2y)^3 \)
Let \( a = x \) and \( b = 2y \):
\[
(x + 2y)^3 = x^3 + 3(x^2)(2y) + 3(x)(2y)^2 + (2y)^3
\]
Calculating each term:
- \( x^3 \)
- \( 3 \cdot x^2 \cdot 2y = 6x^2y \)
- \( 3 \cdot x \cdot (2y)^2 = 3 \cdot x \cdot 4y^2 = 12xy^2 \)
- \( (2y)^3 = 8y^3 \)
Combining these, we get:
\[
(x + 2y)^3 = x^3 + 6x^2y + 12xy^2 + 8y^3
\]
### 26. \( (2a + b)^3 \)
Let \( a = 2a \) and \( b = b \):
\[
(2a + b)^3 = (2a)^3 + 3(2a)^2b + 3(2a)b^2 + b^3
\]
Calculating each term:
- \( (2a)^3 = 8a^3 \)
- \( 3(2a)^2b = 3 \cdot 4a^2b = 12a^2b \)
- \( 3(2a)b^2 = 6ab^2 \)
- \( b^3 \)
Combining these, we get:
\[
(2a + b)^3 = 8a^3 + 12a^2b + 6ab^2 + b^3
\]
### 27. \( (2x - 5)^3 \)
Let \( a = 2x \) and \( b = -5 \):
\[
(2x - 5)^3 = (2x)^3 + 3(2x)^2(-5) + 3(2x)(-5)^2 + (-5)^3
\]
Calculating each term:
- \( (2x)^3 = 8x^3 \)
- \( 3(2x)^2(-5) = -60x^2 \)
- \( 3(2x)(25) = 150x \)
- \( (-5)^3 = -125 \)
Combining these, we get:
\[
(2x - 5)^3 = 8x^3 - 60x^2 + 150x - 125
\]
### 28. \( (a - b)^3 \)
The expression given is incorrect. The correct expansion is:
\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\]
### 29. \( (3a - 5)^3 \)
Let \( a = 3a \) and \( b = -5 \):
\[
(3a - 5)^3 = (3a)^3 + 3(3a)^2(-5) + 3(3a)(-5)^2 + (-5)^3
\]
Calculating each term:
- \( (3a)^3 = 27a^3 \)
- \( 3(3a)^2(-5) = -135a^2 \)
- \( 3(3a)(25) = 225a \)
- \( (-5)^3 = -125 \)
Combining these, we get:
\[
(3a - 5)^3 = 27a^3 - 135a^2 + 225a - 125
\]
### Summary of Results
1. \( (x + 2y)^3 = x^3 + 6x^2y + 12xy^2 + 8y^3 \)
2. \( (2a + b)^3 = 8a^3 + 12a^2b + 6ab^2 + b^3 \
Quick Answer
1. \( (x + 2y)^3 = x^3 + 6x^2y + 12xy^2 + 8y^3 \)
2. \( (2a + b)^3 = 8a^3 + 12a^2b + 6ab^2 + b^3 \)
3. \( (2x - 5)^3 = 8x^3 - 60x^2 + 150x - 125 \)
4. \( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \)
5. \( (3a - 5)^3 = 27a^3 - 135a^2 + 225a - 125 \)
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit