Lyons Ray
06/02/2023 · High School
3) \( (a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
The expression you provided is the expansion of the binomial \((a + b)^3\) using the Binomial Theorem. The Binomial Theorem states that:
\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\]
For \(n = 3\), \(x = a\), and \(y = b\), we can expand \((a + b)^3\) as follows:
\[
(a + b)^3 = \sum_{k=0}^{3} \binom{3}{k} a^{3-k} b^k
\]
Calculating the individual terms:
- For \(k = 0\): \(\binom{3}{0} a^{3} b^{0} = 1 \cdot a^{3} \cdot 1 = a^{3}\)
- For \(k = 1\): \(\binom{3}{1} a^{2} b^{1} = 3 \cdot a^{2} \cdot b = 3a^{2}b\)
- For \(k = 2\): \(\binom{3}{2} a^{1} b^{2} = 3 \cdot a \cdot b^{2} = 3ab^{2}\)
- For \(k = 3\): \(\binom{3}{3} a^{0} b^{3} = 1 \cdot 1 \cdot b^{3} = b^{3}\)
Putting it all together, we have:
\[
(a + b)^3 = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}
\]
Thus, the expression you provided is indeed correct:
\[
(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}
\]
Quick Answer
The expression \((a + b)^{3}\) expands to \(a^{3} + 3a^{2}b + 3ab^{2} + b^{3}\).
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