Problem 14 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Order of the Expansion

The term \((1 + x^3)^3\) indicates that we are expanding a binomial expression raised to the third power. This means we will need to use the fourth row of Pascal's Triangle, which corresponds to the coefficients for \(n = 3\).

Step 2: Determine Coefficients from Pascal's Triangle

For \(n = 3\), the row in Pascal's Triangle is \(1, 3, 3, 1\). These numbers are the coefficients of the expanded terms.

Step 3: Write Down the Structure of the Expansion

Using the coefficients \(1, 3, 3, 1\), the terms start as:- \(1 \cdot (1)^3 \cdot (x^3)^0\) - \(3 \cdot (1)^2 \cdot (x^3)^1\) - \(3 \cdot (1)^1 \cdot (x^3)^2\)- \(1 \cdot (1)^0 \cdot (x^3)^3\) Each term uses a coefficient from Pascal's Triangle, powers of the first term \((1)\), and powers of the second term \((x^3)\).

Step 4: Calculate Each Term in the Expansion

- The first term is \(1 \cdot 1^3 \cdot x^0 = 1\).- The second term is \(3 \cdot 1^2 \cdot x^3 = 3x^3\).- The third term is \(3 \cdot 1^1 \cdot (x^3)^2 = 3x^6\).- The fourth term is \(1 \cdot 1^0 \cdot (x^3)^3 = x^9\).

Step 5: Compile the Complete Expanded Form

Combine all the terms from Step 4 to write the expanded form:\(1 + 3x^3 + 3x^6 + x^9\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a useful tool in mathematics that allows us to expand expressions of the form \((a + b)^n\) where \(n\) is a non-negative integer. This theorem provides a compact way to find all the terms without multiplying the expression repeatedly. The essence of the theorem is that each term in the expansion is made up of coefficients that can be easily determined using Pascal's Triangle.
The general form of the theorem is:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]where \(\binom{n}{k}\) is a binomial coefficient. These coefficients are precisely what you see in Pascal's Triangle.
Here's how the process works:

• Identify the power \(n\) in the binomial expression.
• Use the corresponding row in Pascal's Triangle to find the coefficients.
• Write each term using these coefficients, and the terms of \(a\) and \(b\) raised to appropriate powers.

In our original step-by-step solution, we used this theorem with \((1 + x^3)^3\) to get the expansion with terms weighted by the coefficients from the respective row in Pascal's Triangle.

Polynomial Expansion

Polynomial expansion is the process of transforming a simple polynomial expression raised to a power into a longer, detailed expression. It's essentially breaking down a compound expression like \((1 + x^3)^3\) into a series of simpler terms added together.
This process often reveals the hidden structure within the polynomial and is extremely useful for seeing how individual terms grow as the polynomial's degree increases. To expand a polynomial using something like the binomial theorem:

• Start with identifying the basic unit of repetition (the binomial) in the expression.
• Apply known expansions, like those from Pascal's Triangle, to deduce each term.
• Ensure that you account for each power of the variable in your calculations.

The expanded form of the expression \(1 + 3x^3 + 3x^6 + x^9\) resulted from our polynomial expansion in the original solution.
This expanded version helps to understand the behavior of the polynomial across different input values.

Exponential Expressions

Exponential expressions involve numbers or variables raised to a power. They are a compact way of representing repeated multiplication. In mathematics, these are especially important in modeling processes that grow or decay exponentially, like compound interest or population growth.
When dealing with polynomial expressions like \(1 + x^3)^3\), you are working with exponential expressions as well. Here, the \(x^3\) term is an exponential expression raised further to a power. Understanding this concept helps clarify why and how each term in an expansion behaves as it does:

• Each term in an exponential form retains its base through the expansion.
• The exponent increases in accordance with multiplication rules.
• This builds a new expression through product terms combined with their coefficients.

Taking apart these terms using polynomial expansion reveals the essence of how exponential terms contribute to the overall expression, like in our specific example where the last term is \(x^9\) after the complete expansion.