Problem 19 - Algebra and Trigonometry

Question

Expand each expression using the Binomial Theorem. $$ (x-2)^{6} $$

Step-by-Step Solution

Answer

The expanded form is \(x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64\).

Step 1: Identify the Binomial Theorem

The Binomial Theorem states: \[ (a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{(n-k)} b^k \]

Step 2: Set up the equation

Here, the expression is \((x-2)^6\). Identify \(a\) and \(b\) from the formula where \(a = x\) and \(b = -2\), and \(n = 6\).

Step 3: Apply the Binomial Theorem

Expand the equation using the Binomial Theorem: \[ (x-2)^6 = \sum_{k=0}^{6} {6 \choose k} x^{(6-k)} (-2)^k \]

Step 4: Calculate each term

Calculate each term of the expansion: \[ \begin{align*} \ {6 \choose 0} x^6 (-2)^0 &= 1 \ x^6 \ {6 \choose 1} x^5 (-2)^1 &= 6 \ x^5 \ (-2) = -12 \ x^5 \ \ {6 \choose 2} x^4 (-2)^2 &= 15 \ x^4 \ 4 = 60 \ x^4 \ {6 \choose 3} x^3 (-2)^3 &= 20 \ x^3 \ (-8) = -160 \ x^3 \ \ {6 \choose 4} x^2 (-2)^4 &= 15 \ x^2 \ 16 = 240 \ x^2 \ {6 \choose 5} x^1 (-2)^5 &= 6 \ x^1 \ (-32) = -192 \ x \ {6 \choose 6} x^0 (-2)^6 &= 1 \ 1 \ 64 = 64 \ \ \end{align*} \]

Step 5: Combine all terms

Add all the calculated terms together to get the expanded form: \[ (x-2)^6 = x^6 - 12 x^5 + 60 x^4 - 160 x^3 + 240 x^2 - 192 x + 64 \]

Key Concepts

Binomial Expansion

The binomial expansion is a way to expand expressions that are raised to a power. For instance, \( (x - 2)^6 \) can be expanded using the Binomial Theorem.
The Binomial Theorem formula is \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{(n-k)} b^k \]
Here, you can see that it helps express any binomial expression raised to a power as a sum of terms involving binomial coefficients and the powers of the terms in the binomial expression.
Understanding the Binomial Theorem makes it easier to expand any polynomial when raised to a specific power, rather than multiplying the binomial by itself multiple times.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial in an extended form. This is essential in algebra as it simplifies operations like addition, subtraction, and multiplication.
For example, expanding \( (x - 2)^6 \) transforms it from a compact form into a sum of multiple terms.
The expanded form we get for \( (x - 2)^6 \) is \[ x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64 \]
This detailed breakdown helps in understanding various algebraic properties such as polynomial degrees, leading coefficients, and constant terms.
Overall, polynomial expansion is a fundamental process to ease the manipulation and solution of algebraic equations.

Combinatorics

Combinatorics plays a significant role in algebra, especially in the context of the Binomial Theorem.
Combinatorics deals with counting, arranging, and finding patterns. In the expansion process, binomial coefficients \( \binom{n}{k} \) count the ways to choose \k\ items from \ items.
These coefficients are found using the formula \[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \]
For example, in the term \ 6 \choose 2\ becomes 15.
Understanding combinatorics helps in calculating coefficients quickly and accurately in binomial expansions.
They also guide us in understanding distributions, permutations, and combinations in higher-level mathematics.

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols.
This includes everything from solving equations to understanding structures and patterns.
The binomial expansion is an application of algebra that converts an expression into a polynomial form.
In the given example, expanding \( (x - 2)^6 \) requires knowledge of algebraic manipulation and identities like binomial coefficients.
Through such expansions, we can solve more complex algebraic equations, factor polynomials, and find roots.
It enhances problem-solving skills and deepens the understanding of mathematical relationships and structures.
So, the Binomial Theorem is an indispensable tool in algebra.