Problem 5 - Algebra and Trigonometry

Question

\(1-12\) . Use Pascal's triangle to expand the expression. $$ (x-1)^{5} $$

Step-by-Step Solution

Answer

Expand \((x-1)^5\) to get \(x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1\).

Step 1: Understanding Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. For expansion of \((x-1)^5\), we focus on the 6th row of Pascal's Triangle, since the row index starts at 0. The 6th row is \([1, 5, 10, 10, 5, 1]\).

Step 2: Setting Up the Expansion

The Binomial Theorem states that \((x-a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} (-a)^k\). For our expression \((x-1)^5\), we use \((-1)^k\) instead of \(a\). We'll use the coefficients from the 6th row of Pascal's Triangle and substitute \(a=1\).

Step 3: Applying Coefficients to Terms

Using the coefficients \([1, 5, 10, 10, 5, 1]\), we write out each term as follows: \(1 \cdot x^5 \cdot (-1)^0 + 5 \cdot x^4 \cdot (-1)^1 + 10 \cdot x^3 \cdot (-1)^2 + 10 \cdot x^2 \cdot (-1)^3 + 5 \cdot x^1 \cdot (-1)^4 + 1 \cdot x^0 \cdot (-1)^5\).

Step 4: Evaluate and Simplify Each Term

Calculate each term: \(1x^5 = x^5, -5x^4 = -5x^4, 10x^3, -10x^2, 5x, -1\). This results in the expanded form of the polynomial.

Step 5: Combine All Terms

Now, put all the evaluated terms together: \(x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1\). This is the final expanded form.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra and combinatorics, providing a formula for expanding expressions that are raised to a power. It's especially helpful in expanding binomials, which are expressions with two terms, such as \((x-1)^n\). The theorem states:\[ (x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}a^k \]Here, each term in the expansion involves coefficients known as binomial coefficients, which can be identified using combinatorial methods or Pascal's Triangle. The index \(k\) runs from 0 to \(n\), representing each term in the expansion.The Binomial Theorem is integral to simplifying problems involving polynomials and provides an efficient way to compute the expansion of binomials without having to multiply them repeatedly. Instead, it uses the structure of the coefficients to deliver each term directly.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a polynomial that has been raised to a power as a full expression with all its terms. When you expand \((x-1)^5\), you're essentially undoing the compact notation and writing it out in full.Using Pascal's Triangle, the coefficients are readily available, which makes the task simpler. Every term in the expanded polynomial corresponds to a number from Pascal's Triangle and powers of \(x\) and the other term, following the Binomial Theorem.In our example, this meant using the 6th row of Pascal's Triangle: \([1, 5, 10, 10, 5, 1]\), and associating these to the terms \(x^5, x^4, x^3, x^2, x^1,\) and \(x^0\), allowing a detailed breakdown of the polynomial's formation.

Combinatorial Coefficients

Combinatorial coefficients, also identified as binomial coefficients, are crucial in understanding polynomial expansions under the Binomial Theorem. They appear in the form of \(\binom{n}{k}\), known as "n choose k," and represent the number of combinations of \(n\) items taken \(k\) at a time.These coefficients can be quickly found using Pascal's Triangle, where each number is the sum of the two directly above it. For any binomial expansion \((x+a)^n\), the coefficients are derived from the nth row of Pascal's Triangle.For \((x-1)^5\), using the 6th row \([1, 5, 10, 10, 5, 1]\) ensures that each term in the polynomial has the correct combinatorial coefficient applied, precisely scaling the contribution of the terms.

Algebra

Algebra involves manipulation and solving of equations, where simplification and expansion of polynomials are fundamental skills. Using tools like the Binomial Theorem helps to transform algebraic expressions into simpler, solvable forms.In expanding \((x-1)^5\), algebraic principles direct how terms are assembled and manipulated. This includes:

• Multiplying coefficients from Pascal's Triangle by each term's respective powers of \(x\) and \((-1)\).
• Paying close attention to the signs when negative terms are involved, such as \((-1)^k\) in the expansion.

By mastering these algebraic concepts, students can tackle more complicated expressions with confidence, employing a structured approach to simplify complex problems into manageable tasks.