Problem 40 - Algebra for College Students

Question

Expand each binomial and simplify.\((1-\sqrt{2})^{5}\)

Step-by-Step Solution

Answer

The expanded form is \(31 - 29\sqrt{2}\).

Step 1: Identify the Binomial Expression

The given expression is \((1-\sqrt{2})^{5}\). This is a binomial expression where \(a = 1\) and \(b = -\sqrt{2}\), raised to the power of 5.

Step 2: Use the Binomial Theorem

The binomial theorem is used to expand binomials raised to a power. The formula is: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\].In this case, \(a = 1\), \(b = -\sqrt{2}\), and \(n = 5\).

Step 3: Calculate Binomial Coefficients and Terms

Now calculate each term using the binomial coefficient:- The first term: \(\binom{5}{0}(1)^5(-\sqrt{2})^0 = 1\)- The second term: \(\binom{5}{1}(1)^4(-\sqrt{2})^1 = -5\sqrt{2}\)- The third term: \(\binom{5}{2}(1)^3(-\sqrt{2})^2 = 20\)- The fourth term: \(\binom{5}{3}(1)^2(-\sqrt{2})^3 = -20\sqrt{2}\)- The fifth term: \(\binom{5}{4}(1)^1(-\sqrt{2})^4 = 10\)- The sixth term: \(\binom{5}{5}(1)^0(-\sqrt{2})^5 = -4\sqrt{2}\)

Step 4: Add Up All the Terms

Combine all the calculated terms:\[1 - 5\sqrt{2} + 20 - 20\sqrt{2} + 10 - 4\sqrt{2}\]

Step 5: Simplify the Expression

Combine like terms:- Constant terms: \(1 + 20 + 10 = 31\)- \(-\sqrt{2}\) terms: \(-5\sqrt{2} - 20\sqrt{2} - 4\sqrt{2} = -29\sqrt{2}\)Thus, the expanded expression is:\[31 - 29\sqrt{2}\]

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool that allows us to expand expressions raised to a power. Imagine you have a binomial, which is simply any expression that includes two terms, like \((a + b)^n\). The binomial theorem gives us a formula to expand this without having to multiply the expression by itself repeatedly.

The formula is: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] This formula tells us that we sum several terms, each involving a binomial coefficient, which we'll explore in the next section. This is helpful because it saves time and reduces errors when dealing with high powers. Instead of calculating \((1-\sqrt{2})\) five times, you can jump directly to expansion using the theorem.

The binomial theorem is incredibly useful in calculations, especially in algebra and probability, where binomials appear often. It lays the groundwork for more complex concepts like calculus and series expansions.

Binomial Coefficients

When expanding a binomial using the binomial theorem, one of the key components are the binomial coefficients. These are numbers that determine how each term in the expansion is weighted. In the formula \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), \(\binom{n}{k}\) represents the binomial coefficient.

Binomial coefficients are also known as combination numbers, as they determine how many ways you can choose \(k\) elements from a set of \(n\) elements. Mathematically, they can be calculated using: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(!\) represents factorial, the product of all positive integers up to that number.

In our example \((1-\sqrt{2})^5\), we used binomial coefficients to weigh each term: \(\binom{5}{0}\), \(\binom{5}{1}\), up to \(\binom{5}{5}\). Each coefficient is crucial for calculating the respective terms in the binomial expansion.

Simplifying Expressions

Once you've expanded a binomial expression, the next step is to simplify it. Simplification involves combining like terms to make the expression as concise as possible. In the example of \((1-\sqrt{2})^5\), after using the binomial theorem, we combine similar components to arrive at the simplest form.

Here's a breakdown of this process:

• First, calculate each term in the expansion using the binomial coefficients and identify terms that are similar. This mainly involves looking for common factors, like similar powers of \(\sqrt{2}\).
• Next, sum the constant terms—to simplify, you only need to treat each number without a variable. For instance, combine coefficients to form a single term.
• Similarly, combine the terms that include \(\sqrt{2}\). Group them effectively to consolidate their values. In our example, the terms containing \(-\sqrt{2}\) were combined as \(-5\sqrt{2} - 20\sqrt{2} - 4\sqrt{2} = -29\sqrt{2}\).

Ultimately, the expanded and simplified expression of our example is \(31 - 29\sqrt{2}\). By simplifying, we make it much easier to understand and use for further calculations.