Problem 23 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(y - 4\right)^3 \)

Step-by-Step Solution

Answer

The expansion of the expression \( \left(y - 4\right)^3 \) using the Binomial Theorem is \( y^3 - 12y^2 + 48y - 64 \)

Step 1: Recall the Binomial Theorem

The Binomial theorem states that: \((a + b)^n = a^n + C(n, 1) a^{n-1} b + C(n, 2) a^{n-2} b^2 + ... C(n, n-1) a b^{n-1} + b^n \) where C(n, r) represents the number of combinations of n items taken r at a time.

Step 2: Write the binomial expression using the theorem

Writing the expression \((y - 4)^3\) in the form dictated by the binomial theorem, we get: \((y - 4)^3 = y^3 + C(3, 1) y^{3-1}*(-4) + C(3, 2) y^{3-2}*(-4)^2 + (-4)^3\)

Step 3: Simplify the expression

Simplify the terms and the expression becomes: \(y^3 - 3*y^2*4 + 3*y*16 - 64\) which simplifies further to: \(y^3 - 12y^2 + 48y - 64\).

Key Concepts

Combinations

In the Binomial Theorem, combinations play a crucial role in determining the coefficients of the expanded terms. When we talk about combinations in mathematics, we refer to the selection of items from a larger pool, where the order does not matter.

In our exercise, we have the expression \((y - 4)^3\). Here, we need to find the coefficients for each term in the expansion, which is done using the combination formula \(C(n, r)\). This notation represents the number of ways to choose \(r\) objects from \(n\) without regard to order. This is calculated as:

• \(C(n, r) = \frac{n!}{r!(n-r)!}\)

For our specific problem with \(n = 3\):

• \(C(3, 0) = 1\)
• \(C(3, 1) = 3\)
• \(C(3, 2) = 3\)
• \(C(3, 3) = 1\)

These numbers are used as coefficients in the binomial expansion, demonstrating how combinations allow us to systematically expand binomial expressions.

Expansion

The concept of expansion in the Binomial Theorem is all about transforming a binomial expression into a polynomial form. The theorem provides a reliable method for expanding expressions of the form \((a + b)^n\). In our exercise, we have a special case with \((y - 4)^3\).

First, recognize the pattern from the theorem:

• \(a^n\)
• \(C(n, 1) a^{n-1} b\)
• \(C(n, 2) a^{n-2} b^2\)
• ...and so on until \(b^n\)

Applying this so-called expansion rule to \((y - 4)^3\):

• First term: \(y^3\)
• Second term: \(C(3, 1) y^{2} (-4) = 3y^2(-4)\)
• Third term: \(C(3, 2) y (-4)^2 = 3y(16)\)
• Fourth term: \((-4)^3 = -64\)

These terms are then combined to form the expanded polynomial expression, which is a critical step in understanding how binomials can be rewritten in a more comprehensive polynomial form.

Simplification

Once the binomial expression is expanded, the next step is simplification, which involves making the polynomial more readable and straightforward. Simplification is about combining like terms, performing any arithmetic operations, and presenting the polynomial in a clean format.

From our expanded form, \(y^3 - 3y^2 \cdot 4 + 3y \cdot 16 - 64\), the goal is to perform arithmetic operations:

• Multiply coefficients and simplify each term: \(y^3 - 12y^2 + 48y - 64\).

This polynomial represents the simplest form of the expanded expression, maintaining the same values but organized more neatly. Simplification removes complexities and prepares the polynomial for further mathematical operations, if necessary.

Simplifying expressions not only aids in clarity but also provides a deeper understanding of the relationships between terms and their coefficients, ensuring a better grasp of polynomial functions.