Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (y-3)4
589
views
10. Combinatorics & Probability
Combinatorics
4:54 minutesProblem 11
Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form.
Verified step by step guidance1
Identify the binomial expression to expand: \((3x + y)^3\).
Recall the Binomial Theorem formula: \(\displaystyle (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.
Set \(a = 3x\), \(b = y\), and \(n = 3\). Write out each term of the expansion using the formula: \(\binom{3}{k} (3x)^{3-k} y^k\) for \(k = 0, 1, 2, 3\).
Calculate each binomial coefficient: \(\binom{3}{0}\), \(\binom{3}{1}\), \(\binom{3}{2}\), and \(\binom{3}{3}\), and simplify the powers of \$3x\( and \)y$ accordingly.
Write the full expanded expression by summing all terms and simplify each term by multiplying coefficients and combining like factors.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Theorem
The Binomial Theorem provides a formula to expand expressions raised to a power, such as (a + b)^n. It states that (a + b)^n equals the sum of terms involving binomial coefficients, powers of a, and powers of b. This theorem simplifies the expansion process without multiplying the binomial repeatedly.
Recommended video:
Guided course
03:41
Special Products - Cube Formulas
Binomial Coefficients
Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n. They appear as coefficients in the expanded form of (a + b)^n and can be found using Pascal's Triangle or the formula C(n, k) = n! / (k!(n-k)!).
Recommended video:
Guided course
03:41Special Products - Cube Formulas
Exponent Rules and Simplification
When expanding binomials, powers of each term must be calculated using exponent rules, such as (x^m)(x^n) = x^(m+n). After expansion, like terms should be combined and simplified to express the result in its simplest form, ensuring clarity and correctness.
Recommended video:
Guided course
7:39Introduction to Exponent Rules
Related Videos
Related Practice