Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand and simplify the expression. \(\left(3 x^{3}-y\right)^{6}\)

Step-by-Step Solution

Answer

The expanded form of the binomial expression \((3 x^{3}-y)^{6}\) is \(729x^{18} -1458x^{15}y + 1215x^{12}y^{2} - 540x^{9}y^{3} + 135x^{6}y^{4} - 18x^{3}y^{5} + y^{6}\)

Step 1: Understanding the Binomial Theorem

The Binomial Theorem is represented as \((a+b)^n=\sum_{k=0}^{n} {n\choose k} a^{n-k}b^{k}\) where \(n\choose k\)=nCk is the combination and k takes all integer values from 0 to n. Here in the given expression \((3x^3-y)^6\), 'a' is '3x^3', 'b' is '-y' and 'n' is '6'.

Step 2: Applying the Binomial Theorem

Applying the binomial theorem, the expanded form of \((3x^3-y)^6\) will be a series of 7 terms (since there's always one more term than the power). The general term looks like \({6\choose k} (3x^{3})^{6-k} (-y)^{k}\).

Step 3: Calculation of terms

Let’s calculate these seven terms individually: \n \Term 1 when k=0: \( {6\choose 0} (3x^{3})^{6} (-y)^{0} = 1 * (729x^{18}) * 1 = 729x^{18}\), \n \Term 2 when k=1: \( {6\choose 1} (3x^{3})^{5} (-y)^{1} = 6 * (243x^{15}) * (-y) = -1458x^{15}y\), \n \Term 3 when k=2: \( {6\choose 2} (3x^{3})^{4} (-y)^{2} = 15 * (81x^{12}) *(y^{2}) = 1215x^{12}y^{2}\), \n \and continue the same for k=3, 4, 5, 6.

Step 4: Summing up all terms

Sum of all these Seven terms, \(729x^{18} -1458x^{15}y + 1215x^{12}y^{2} - 540x^{9}y^{3} + 135x^{6}y^{4} - 18x^{3}y^{5} + y^{6}\), gives the expanded form of the given binomial expression.

Key Concepts

Combination

In the world of mathematics, a combination is a selection of items where the order does not matter. It typically forms a part of the process when expanding expressions using the Binomial Theorem. To sound more mathematical, it's expressed as \( \binom{n}{k} \).
Here, \( n \) represents the total number of items, and \( k \) is the number of items to choose. The formula for a combination is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \(!\) denotes factorial, meaning you multiply all positive integers up to that number.

When we expand a binomial expression like \((3x^3 - y)^6\), combinations help us figure out the coefficients for each term. For example, in this expansion, terms start with coefficients derived from \( \binom{6}{k} \), where \( k \) ranges from 0 to 6. These coefficients play a crucial role in forming the terms of our polynomial expansion.

Polynomial Expansion

Polynomial expansion is a process that involves expressing a binomial expression as a sum of terms. Each term is a product of coefficients, powers of variables, and possibly negative signs if one of the binomial terms is negative.
Using the Binomial Theorem, we can expand expressions of the form \((a + b)^n\).
This theorem tells us how to derive each term of the expansion. The expression can be written as a sum, \( \sum_{k=0}^{n} {n\choose k} a^{n-k} b^k \). Applying this to \((3x^3 - y)^6\):

• Each term of the expansion takes the form \( {6\choose k}\cdot(3x^{3})^{6-k} \cdot (-y)^{k} \).
• Because '-y' is negative, the sign of each term will alternate based on the power of \(-y\).
• There are seven terms from \( k = 0 \) to \( k = 6 \), reflecting the powers of the terms in the binomial expansion.

Exponentiation

Exponentiation refers to the operation of raising a number or expression to a certain power. In the realm of the Binomial Theorem, powers are both essential and handy.
For the expression \((3x^3 - y)^6\), we use exponentiation to calculate the powers of terms within the polynomial expansion. Key points about exponentiation in this context:

• In each term, the part \((3x^3)^{6-k}\) involves raising \(3x^3\) to the power \(6-k\). For example, when \( k = 0 \), this portion becomes \(9x^{18}\), achieved by calculating the cube of 3 and adjusting the exponent of x accordingly.
• Similarly, \((-y)^k\) dictates the power of \(-y\) in each term, contributing to the sign alternation among terms due to the negative value.

Exponentiation is crucial for defining the structure of each term in the expansion.

Mathematical Expression Simplification

Simplification in mathematics often involves restructuring an expression to make it easier to understand or work with. Here, with the Binomial Theorem expansion, we end up with complex expressions that need simplifying.
Simplification involves calculating powers, multiplying coefficients, and combining like terms. In the context of \((3x^3 - y)^6\):

• You compute each coefficient using combinations like \(\binom{6}{k}\) and multiplying them by any factors like \(3^{6-k}\).
• Each term in the polynomial form needs multiplying by powers and possible negative signs, then the terms are summed to provide the final simplified expression.
• The ultimate goal is to arrive at an expression involving clear, concise terms like \[729x^{18} -1458x^{15}y + 1215x^{12}y^{2} - 540x^{9}y^{3} + 135x^{6}y^{4} - 18x^{3}y^{5} + y^{6}\].

Keeping the terms organized and coherent aids vastly in grasping the overall expansion.