Problem 81 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand and then simplify the result: \(\left(x^{2}+x+1\right)^{3}\) Hint: Write \(x^{2}+x+1\) as \(x^{2}+(x+1)\)

Step-by-Step Solution

Answer

The expansion of \( (x^{2}+(x+1))^{3}\) according to the binomial theorem is \(x^{6}+3x^{4}+3x^{2}+1\)

Step 1: Identify the Binomial

Identify the binomial in the given expression which is \(x^{2}+x+1\), this can be rewritten as \(x^{2}+(x+1)\)

Step 2: Apply Binomial Theorem

To expand \( (x^{2}+(x+1))^3 \), apply the binomial theorem. This results in a sum of terms where each term is a product of a binomial coefficient and the first term raised to a power as well as the second term raised to a power. Using the binomial theorem, we get \({3\choose 0}x^{2*(3-0)}(x+1)^{0}+ {3\choose 1}x^{2*(3-1)}(x+1)^{1}+ {3\choose 2}x^{2*(3-2)}(x+1)^{2}+ {3\choose 3}x^{2*(3-3)}(x+1)^{3}\) which simplifies to \(x^{6}+3x^{4}+3x^{2}+1\)

Step 3: Simplify the Result

Simplify the expanded binomial further if possible, in this case no further simplification is required, so the answer is \(x^{6}+3x^{4}+3x^{2}+1\)

Key Concepts

Binomial Expansion

When you encounter a binomial raised to a power, such as \( (a+b)^n \), binomial expansion allows you to express this expression as a sum of terms involving coefficients, the first term of the binomial, and the second term of the binomial, each raised to a power. The Binomial Theorem, a pivotal concept in algebra, comes into play here. Let's breakdown how you can use it.

Imagine you are tasked to expand \( (x^2 + x + 1)^3 \). Initially, it may seem daunting, but with the theorem, it becomes more manageable. The key here is to identify the binomial suitable for expansion, which is \( x^2 + x + 1 \), but cleverly rephrased as \( x^2 + (x + 1) \). Apply the theorem by starting with the power of the binomial \( n=3 \) in this exercise, and then lay out each term using binomial coefficients.

In a step-by-step approach, you'll see a sequence where each term consists of a binomial coefficient—represented using the 'choose' notation like \( {n\choose k} \)—multiplied by the successive powers of the first term, here \( x^2 \), decrementing from \( n \) down to zero, and the second term \( x + 1 \) raised from zero to \( n \). Each term collectively represents a critical piece of the expanded polynomial.

Binomial Coefficient

The coefficients you see in the binomial expansion are known as binomial coefficients. These numbers are not just ordinary numbers but hold a significant position in combinatorics, representing the number of ways to choose 'k' elements out of a set of 'n' elements, written as \( {n\choose k} \), also known as 'n choose k'.

In the context of the given problem, to expand \( (x^2 + x + 1)^3 \), binomial coefficients can be visualized as the numbers that precede each term in the expansion. The way to calculate them involves factorials, as \( {n\choose k} = \frac{n!}{k!(n-k)!} \), where \( ! \) denotes the factorial of a number, meaning the product of all positive integers up to that number.

For instance, \( {3\choose 0} \) is 1, \( {3\choose 1} \) is 3, and so on. These coefficients multiplicatively support each term in the polynomial expansion, revealing how combinatorics intertwines with algebra, particularly in expanding binomials.

Polynomial Simplification

After expanding a binomial using the Binomial Theorem, the next step is often to simplify the resultant polynomial. Simplification might involve combining like terms, which are terms that have the same variables raised to the same powers, or it could just be a matter of expressing the polynomial in a standard form.

In the exercise \( (x^2 + x + 1)^3 \), the result of the binomial expansion is \( x^6 + 3x^4 + 3x^2 + 1 \). In this scenario, the simplification process is relatively straightforward since there are no like terms to combine—the simplification involves simply writing out the terms of the expanded polynomial in descending powers of \( x \). The result is already in its simplest form.

Simplification plays a crucial role in ensuring the clarity and ease of understanding of algebraic expressions. It's the final touch in the binomial expansion process, making complex expressions more accessible and manageable, whether for further mathematical operations or just for interpretation.