Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand and simplify the expression. \((4 x-3 y)^{4}\)

Step-by-Step Solution

Answer

The simplified expression is \(256x^4 - 768x^3y + 864x^2y^2 - 432xy^3 + 81y^4\)

Step 1: Identify 'a', 'b' and 'n'

In the given expression \((4x-3y)^4\), a is \(4x\), b is \(-3y\) (not just \(3y\)), and n is 4.

Step 2: Expansion using the Binomial Theorem

Expand the expression using the Binomial theorem formula \(\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\). By substituting \(a=4x\), \(b=-3y\), and \(n=4\), we get: \(\binom{4}{0}(4x)^4(-3y)^0 + \binom{4}{1}(4x)^3(-3y)^1 + \binom{4}{2}(4x)^2(-3y)^2 + \binom{4}{3}(4x)^1(-3y)^3 + \binom{4}{4}(4x)^0(-3y)^4\)

Step 3: Calculate binomial coefficients and simplify

Calculate the binomial coefficients and simplify each term: \(1*(256x^4) + 4*(64x^3)*(-3y) + 6*(16x^2)*(9y^2) + 4*(4x)*(-27y^3) + 1*(-81y^4)\). When simplified this becomes: \(256x^4 - 768x^3y + 864x^2y^2 - 432xy^3 + 81y^4\)

Key Concepts

Binomial Coefficients

The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions raised to a power. One crucial element in this theorem is the binomial coefficients. These coefficients are represented by \( \binom{n}{k} \), which is read as \(n\) choose \(k\). This notation indicates the number of ways to choose \(k\) items from \(n\) available options.

• The binomial coefficient \( \binom{n}{k} \) is calculated using the formula: \( \frac{n!}{k!(n-k)!} \).
• In our example, consider \((4x-3y)^4\). Here, our values are \(n=4\) and \(k\) varies from 0 to 4 as we expand.
• Therefore, calculating the coefficients involves using \(\binom{4}{0}\), \(\binom{4}{1}\), \(\binom{4}{2}\), \(\binom{4}{3}\), and \(\binom{4}{4}\).

Understanding and using these coefficients is key to simplifying and completing the expansion correctly.

Polynomial Expansion

Expanding a polynomial using the Binomial Theorem involves breaking down the expression and multiplying it out to reveal its individual terms. The binomial expression \((4x-3y)^4\) can be expanded with the formula \( \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k \). Each term in the expansion is a product of binomial coefficients and powers of the expressions involved.

• The fullness of a polynomial's expansion can be predicted by the degree of the polynomial, in this case, 4.
• The expansion produces five terms: \[\binom{4}{0}(4x)^4(-3y)^0, \binom{4}{1}(4x)^3(-3y)^1, \binom{4}{2}(4x)^2(-3y)^2, \binom{4}{3}(4x)^1(-3y)^3, \binom{4}{4}(4x)^0(-3y)^4\].
• It's important to keep track of each term's sign, especially when \(b\) is negative.

This systematic method ensures that no terms are lost and that all are properly accounted for and calculated.

Algebraic Simplification

Once the polynomial expansion is complete, the next step is algebraic simplification. Simplification involves performing the arithmetic operations and reducing the expression to its most concise form while ensuring accuracy. In the example \((4x-3y)^4\), the expansion results in several terms, each needing calculation:

• \(256x^4\) from \(1 \times 256x^4\)
• \(-768x^3y\) from \(4 \times 64x^3 \times -3y\)
• \(864x^2y^2\) from \(6 \times 16x^2 \times 9y^2\)
• \(-432xy^3\) from \(4 \times 4x \times -27y^3\)
• \(81y^4\) from \(1 \times -81y^4\)

These components, when combined, give the final result: \(256x^4 - 768x^3y + 864x^2y^2 - 432xy^3 + 81y^4\).
This step not only helps in reaching a solution but also in fundamental arithmetic reinforcement.

Mathematical Notation

Mathematical notation is a language designed to describe mathematical ideas using symbols and numbers efficiently. It allows us to capture complex operations in concise phrases. In binomial expansions, several notations become particularly impactful.

• The expression \((4x-3y)^4\) uses exponentiation to denote a repeated application of a factor.
• \( \binom{n}{k} \) is a compact notation for expressing binomial coefficients.
• Variables like \(x\) and \(y\) represent unknown or variable quantities, while constants give exact values.

Grasping this notation is crucial since it allows mathematicians and students alike to read and interpret calculations swiftly and without ambiguity. Understanding and correctly applying this notation is vital for all mathematical work, ensuring clear communication and logical progression through problems.