Problem 26 - College Algebra

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x-2)^{5} $$

Step-by-Step Solution

Answer

\(x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32\)

Step 1: Introduction and Setup of the Binomial Theorem

The Binomial Theorem states that for any numbers \(a\) and \(b\), and a positive integer \(n\), the expansion of \( (a + b)^n \) is given by \( (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r} \) where \( \binom{n}{r} \) are the binomial coefficients from Pascal's triangle or computed as \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) where `!` denotes factorial. The expression to be expanded here is \( (x-2)^5 \). This is of the same form as \( (a + b)^n \), by treating \(x\) as \(a\), \(-2\) as \(b\) and \(5\) as \(n\)

Step 2: Implementation of the Binomial Theorem

Following the formula, the terms of the expanded binomial can be computed as follows: \( \binom{5}{0} x^5 (-2)^0 \), \( \binom{5}{1} x^4 (-2)^1 \), \( \binom{5}{2} x^3 (-2)^2 \), \( \binom{5}{3} x^2 (-2)^3 \), \( \binom{5}{4} x^1 (-2)^4 \), \( \binom{5}{5} x^0 (-2)^5 \)

Step 3: Calculating the Binomial Coefficients and Simplifying

Calculating the binomial coefficients yields: \(1\cdot x^5 \cdot (-2)^0\), \(-5\cdot x^4 \cdot (-2)^1\), \(10 \cdot x^3 \cdot (-2)^2\), \(-10 \cdot x^2 \cdot (-2)^3\), \(5 \cdot x \cdot (-2)^4\), \(-1 \cdot (-2)^5\). Simplifying each expression using the properties of powers gives: \(x^5\), \(-10 x^4\), \(40 x^3\), \(-80 x^2\), \(80 x\), \(-32\).

Step 4: Writing the Final Result

The expanded form of the binomial is then the sum of the terms calculated above, which gives the final answer: \(x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32\)

Key Concepts

Simplifying Binomial Expressions

Understanding how to simplify a binomial expression can unlock a world of polynomial algebra. When dealing with an expression such as \( (x-2)^{5} \), simplifying means expanding it into a readable polynomial form. This process involves two key players: the binomial coefficients and the exponents of both terms in the binomial.

The Binomial Theorem is a powerful tool for simplifying such expressions, breaking them down into a series of terms that involve powers of both binomial terms and the binomial coefficients. Taking our example of \( (x-2)^{5} \), we apply the theorem step by step. Initially, we consider each term where \( x \) is raised to a decremented power and \( -2 \) to an incremented power, from \( (x)^{5}(-2)^{0} \) through to \( (x)^{0}(-2)^{5} \).

Remember, simplification is all about clarity and conciseness, so once the expansion is done, we simplify by calculating each term and collecting like terms. In this case, we get a polynomial in descending order of powers of x, which is much easier to interpret and use in further computations or graphing.

Pascal's Triangle

Pascal's triangle is not just a pattern of numbers; it's a goldmine for combinatorics and algebra. Each row of Pascal's triangle provides the coefficients for the expansion of a binomial expression raised to a power equivalent to the row number.

For instance, the fifth row of Pascal's triangle \(1, 5, 10, 10, 5, 1\) directly gives us the coefficients needed for the expansion of \( (x-2)^{5} \). These numbers are essential because they tell us how many combinations of terms we get when we distribute the binomial.

Implementing Pascal's Triangle

To practically apply Pascal's Triangle to our binomial expression, we simply pick the appropriate row that aligns with the power of the binomial and apply each number as a coefficient to the expanded terms. This approach can drastically simplify the process of finding the coefficients without delving into factorial notation, thereby streamlining our efforts to expand binomial expressions.

Factorial Notation

Factorial notation is the exclamation point of mathematics; it signifies the product of an integer and all the integers below it down to one. It is denoted by '!' and is a shorthand to express large multiplications quickly and efficiently.

When working with the Binomial Theorem, factorials appear in the computation of binomial coefficients. The coefficient \( \binom{n}{r} \) is the number of ways to choose \( r \) elements from a set of \( n \) distinct elements, and it's calculated using factorials: \( \frac{n!}{r!(n-r)!} \).

Understanding factorial notation is vital for simplifying binomial expressions as it allows us to compute binomial coefficients accurately. In the example of \( (x-2)^{5} \), we saw coefficients such as \( \binom{5}{1} \) which translates to \( \frac{5!}{1!(5-1)!} = 5 \).

By grasping the concept of factorial notation, students can confidently expand binomials and deal with larger expressions where manual counting of combinations would be impractical. Incidentally, factorial notation is also a building block for understanding permutations and combinations in probability and statistics, making it a concept with broad applications in mathematics.