Problem 33 - College Algebra with Modeling and Visualization

Question

Use Pascal's triangle to help expand the expression. $$ (3 x+1)^{4} $$

Step-by-Step Solution

Answer

The expanded form is \(81x^4 + 108x^3 + 54x^2 + 12x + 1\).

Step 1: Understand The Binomial Theorem

Before we use Pascal's Triangle, it's important to recall the Binomial Theorem, which allows us to expand expressions of the form \((a+b)^n\). The theorem states:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]where \(\binom{n}{k}\) are the binomial coefficients.

Step 2: Identify "a" and "b" in the Expression

For the expression \((3x + 1)^4\), identify "a" and "b" where \(a = 3x\) and \(b = 1\). We need to expand this expression using the Binomial Theorem with \(n = 4\).

Step 3: Determine the Coefficients Using Pascal's Triangle

Pascal's Triangle provides the coefficients for binomial expansions. For the 4th row (since our exponent is 4), Pascal's Triangle gives the coefficients: 1, 4, 6, 4, 1.

Step 4: Expand the Expression

Using the coefficients from Pascal's Triangle and the values of "a" and "b" identified earlier:\[(3x + 1)^4 = 1 \cdot (3x)^4 + 4 \cdot (3x)^3 \cdot 1 + 6 \cdot (3x)^2 \cdot 1^2 + 4 \cdot (3x)^1 \cdot 1^3 + 1 \cdot 1^4\]Calculate each term.

Step 5: Calculate Each Term

- First term: \(1 \cdot (3x)^4 = 81x^4\)- Second term: \(4 \cdot (3x)^3 \cdot 1 = 108x^3\)- Third term: \(6 \cdot (3x)^2 \cdot 1^2 = 54x^2\)- Fourth term: \(4 \cdot (3x)^1 \cdot 1^3 = 12x\)- Fifth term: \(1 \cdot 1^4 = 1\)

Step 6: Write the Expanded Form

Combine all the calculated terms to write the expanded expression:\[81x^4 + 108x^3 + 54x^2 + 12x + 1\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that allows us to expand expressions raised to a power, usually in the form of \((a + b)^n\). The theorem is very powerful because it provides a method for expanding the expression into a sum of terms involving binomial coefficients. These coefficients can be determined using a formula or visually represented in Pascal's Triangle.
The theorem states that for any non-negative integer \(n\): \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]Each term of the expansion consists of a coefficient \(\binom{n}{k}\), which is the binomial coefficient, alongside powers of \(a\) and \(b\), which decrease and increase, respectively, as you progress through the terms.

• \(a\) and \(b\) are any numbers or algebraic expressions.
• \(n\) is a non-negative integer representing the power to which the binomial is raised.
• \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements, also known as a binomial coefficient.

Understanding this theorem is crucial in simplifying and solving problems involving polynomial expansion, which can be a common task in many mathematical and applied science problems.

Binomial Expansion

The term 'binomial expansion' refers to the process of expanding a binomial expression like \((a + b)^n\) into a polynomial. By following the principles of the binomial theorem, each term in the expanded polynomial can be calculated easily, often using the coefficients provided by Pascal's Triangle.
To perform a binomial expansion for any given power \(n\):

• Begin by identifying your values for \(a\) and \(b\); these can be numbers or algebraic expressions.
• Determine the appropriate coefficients by using Pascal's Triangle or the binomial coefficients formula \(\binom{n}{k}\).
• Expand the expression by calculating each term using \(a^{n-k} b^k\) adjusted by its respective coefficient.

A practical example is expanding \((3x + 1)^4\), which involves identifying \(a = 3x\) and \(b = 1\), and using the 4th row of Pascal's Triangle for the coefficients: 1, 4, 6, 4, 1. Following these steps, the expansion yields a polynomial that illustrates the simplicity and power of the binomial theorem when dealing with polynomials.

Binomial Coefficients

Binomial coefficients are a key component in the binomial theorem. They determine the weight of each term in the binomial expansion and are represented by \(\binom{n}{k}\), which is read as "n choose k." This represents the number of ways to choose \(k\) items from a set of \(n\) items, a concept often used in combinatorics.
These coefficients can be calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \(n!\) (read n factorial) is the product of all positive integers up to \(n\).
Alternatively, they can be visually identified using Pascal's Triangle. Pascal’s Triangle is a triangular array where each row represents the coefficients for the binomial expansion of \((a + b)^n\). For example, the 4th row of Pascal’s Triangle is 1, 4, 6, 4, 1, corresponding to the expansion coefficients of \((a+b)^4\).

• Easy visualization using Pascal’s Triangle makes calculating binomial coefficients simpler, especially for smaller values of \(n\).
• For larger values of \(n\), the factorial method may be preferred to compute the coefficients.

Using these coefficients in a binomial expansion simplifies the process and links algebraic operations to combinatorics, bridging fundamental mathematical concepts.