Problem 14 - Precalculus: Mathematics for Calculus

Question

Pascal's Triangle Use Pascal's triangle to expand the expression. $$\left(1+x^{3}\right)^{3}$$

Step-by-Step Solution

Answer

The expansion is \(1 + 3x^3 + 3x^6 + x^9\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\). We will use this theorem to expand \((1 + x^3)^3\).

Step 2: Identify the Power and Coefficients

Identify \(a = 1\), \(b = x^3\), and \(n = 3\) from the expression \((1 + x^3)^3\). We will use the coefficients from the 4th row of Pascal's Triangle, which are 1, 3, 3, 1.

Step 3: Apply Coefficients from Pascal's Triangle

The binomial coefficients for \(n = 3\) are 1, 3, 3, 1. Use these coefficients to form the expansion terms: \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), \(\binom{3}{3} = 1\).

Step 4: Expand the Expression

Using the coefficients and the Binomial Theorem: \[ (1+x^3)^3 = \binom{3}{0}(1)^3(x^3)^0 + \binom{3}{1}(1)^2(x^3)^1 + \binom{3}{2}(1)^1(x^3)^2 + \binom{3}{3}(1)^0(x^3)^3 \].

Step 5: Simplify Each Term

Calculate each term: - \(\binom{3}{0}(1)^3(x^3)^0 = 1\)- \(\binom{3}{1}(1)^2(x^3)^1 = 3x^3\)- \(\binom{3}{2}(1)^1(x^3)^2 = 3x^6\)- \(\binom{3}{3}(1)^0(x^3)^3 = x^9\)

Step 6: Write the Final Expanded Expression

Combine all the simplified terms: \((1 + x^3)^3 = 1 + 3x^3 + 3x^6 + x^9\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions raised to a power. It provides a way to break down expressions like \((a + b)^n\) into simpler terms. The theorem states that:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here, the binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from \(n\) items and is found using Pascal's Triangle. The \texttt{sum} sign \(\sum\) indicates that you'll sum up all the products of these coefficients with the respective terms.
For example, if you need to expand \((1 + x)^3\), you identify it as \(a = 1\), \(b = x\), and \(n = 3\). You then use coefficients from the 4th row of Pascal's Triangle, which are 1, 3, 3, and 1. This gives you the terms:

• \((1)^3 \cdot (x)^0\)
• \((1)^2 \cdot (x)^1\)
• \((1)^1 \cdot (x)^2\)
• \((1)^0 \cdot (x)^3\)

Combine these to get \((1 + x)^3 = 1 + 3x + 3x^2 + x^3\). By applying this theorem, expanding even complex polynomials becomes systematic and straightforward.

Binomial Expansion

Binomial Expansion specifically refers to the actual process of using the Binomial Theorem to expand expressions. In our example, the expression \((1 + x^3)^3\) is the binomial form you want to expand. The components \(1\) and \(x^3\) replace the \(a\) and \(b\) terms of the general theorem.
You use the coefficients from Pascal's Triangle that correspond to \(n = \text{exponent}\). So, when the exponent is \(3\), the 4th row's coefficients are \(1, 3, 3, 1\) because Pascal's Triangle starts counting at zero.
Once you have your coefficients, you plug them into the theorem formula. Apply each coefficient to its respective term and power:

• First term: \(1 \cdot (1)^3 \cdot (x^3)^0 = 1\)
• Second term: \(3 \cdot (1)^2 \cdot (x^3)^1 = 3x^3\)
• Third term: \(3 \cdot (1)^1 \cdot (x^3)^2 = 3x^6\)
• Fourth term: \(1 \cdot (1)^0 \cdot (x^3)^3 = x^9\)

Add these results together, and you get \((1 + x^3)^3 = 1 + 3x^3 + 3x^6 + x^9\). Remember, the binomial expansion simplifies complex algebraic tasks to systematic steps using known coefficients.

Polynomial Expansion

Polynomial Expansion is a broader term that includes Binomial Expansion, where expressions of one variable raised to a power can be broken into simpler pieces. A polynomial is any algebraic expression composed of variables and coefficients, which can be handled quite efficiently through expansions.
Using Pascal's Triangle, the pattern is easy to spot, especially when dealing with binomial expressions. For instance, the triangle provides the coefficients needed when expanding binomials of the form \((a + b)^n\). However, when talking about polynomial expansion, this refers to dealing with multiple variables and more complex expressions, which might not strictly adhere to the binomial form.
Regardless of complexity, simplifying or expanding involves breaking down the expression piece by piece. Key principles and tools remain the same:

• Identify the terms \(a\) and \(b\).
• Find the associated coefficients through Pascal’s Triangle or combinations.
• Use step-by-step multiplication to expand each component of the polynomial.

Ultimately, the practice of polynomial expansion will arm you with a versatile skill—whether it's expanding binomial expressions or tackling more elaborate ones, understanding these basics ensures clarity in any level of algebraic manipulation.