Problem 44 - College Algebra with Modeling and Visualization

Question

Use Pascal's triangle to help expand the expression. $$ \left(3 x^{2}+y^{3}\right)^{4} $$

Step-by-Step Solution

Answer

The expanded expression is \(81x^8 + 108x^6y^3 + 54x^4y^6 + 12x^2y^9 + y^{12}\).

Step 1: Identify the Binomial Components

Recognize that the expression \((3x^2 + y^3)^4\) is a binomial of the form \((a + b)^n\) with \(a = 3x^2\), \(b = y^3\), and \(n = 4\).

Step 2: Use Pascal's Triangle to Find Coefficients

Draw the 4th row of Pascal's triangle, which corresponds to \(n = 4\). The row is: \(1, 4, 6, 4, 1\). These numbers will be used as coefficients in the binomial expansion.

Step 3: Expand Using Binomial Theorem

According to the binomial theorem, the expansion will be \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. Use the coefficients from Pascal's triangle and substitute \(a = 3x^2\), \(b = y^3\), \(n = 4\):1. \(a^4 = (3x^2)^4\), \(b^0 = 1\), coefficient 1: \[ 1 \times (3x^2)^4 \times 1 = 81x^8 \]2. \(a^3 = (3x^2)^3\), \(b^1 = y^3\), coefficient 4: \[ 4 \times (3x^2)^3 \times y^3 = 108x^6y^3 \]3. \(a^2 = (3x^2)^2\), \(b^2 = (y^3)^2\), coefficient 6: \[ 6 \times (3x^2)^2 \times y^6 = 54x^4y^6 \]4. \(a^1 = (3x^2)\), \(b^3 = (y^3)^3\), coefficient 4: \[ 4 \times (3x^2) \times y^9 = 12x^2y^9 \]5. \(a^0 = 1\), \(b^4 = (y^3)^4\), coefficient 1: \[ 1 \times 1 \times y^{12} = y^{12} \]

Step 4: Combine All Terms

Compile all calculated terms to form the expanded expression:\[81x^8 + 108x^6y^3 + 54x^4y^6 + 12x^2y^9 + y^{12}\].

Key Concepts

Pascal's triangle and its role in binomial expansion

Pascal's triangle is a simple yet powerful tool in algebra, particularly for expanding expressions raised to a power. It is named after the French mathematician Blaise Pascal and is a triangular array of numbers. Each row represents the coefficients of the binomial expansion for a given power.
Pascal's triangle begins with a single number "1" at the top. Each subsequent number in the triangle is the sum of the two numbers directly above it from the previous row.

• The first row is 1.
• The second row is 1, 1.
• The third row is 1, 2, 1.
• The fourth row is 1, 3, 3, 1, and so on.

In the exercise provided, the expression \( (3x^2 + y^3)^4 \) was expanded. For this, the fourth row of Pascal's triangle (1, 4, 6, 4, 1) was used as the coefficients for the binomial terms. This approach simplifies finding the coefficients without directly calculating the binomial coefficients using the formula.

Understanding the binomial theorem

The binomial theorem provides a way to express the powers of binomials as a sum of terms involving coefficients, powers of the first term, and powers of the second term. It is expressed as: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] Here, \( \binom{n}{k} \) are binomial coefficients that can be found in the corresponding row of Pascal's triangle.
The process is simple:

• Identify the terms (\(a\) and \(b\)) and the exponent \(n\).
• Write out the powers for \(a\) starting from \(n\) down to 0 and \(b\) starting from 0 up to \(n\).
• Apply the coefficients from Pascal's triangle.

This results in a series of terms that, when combined, give the expanded expression for the original binomial raised to a power. In the exercise, this method was used with \(a = 3x^2\), \(b = y^3\), and \(n = 4\). This process facilitates the organization of different terms allowing each to be clearly calculated and combined into the final expanded form.

What are algebraic expressions?

Algebraic expressions are mathematical phrases that can contain numbers, variables (like \(x\) or \(y\)), and operations (such as addition, subtraction, multiplication, and division). They are the language of algebra and form the basis for solving equations and understanding mathematical relationships.
In the context of binomial expansion, an expression like \(3x^2 + y^3\) is considered a binomial because it has two terms. Here:

• \(3x^2\) is a term with a coefficient of 3 and includes the variable \(x\) squared.
• \(y^3\) is a term with \(y\) raised to the third power.

In algebra, expanding such expressions involves using established methods like the binomial theorem, allowing us to manipulate and simplify them. This expansion process helps in finding specific terms, making calculations in polynomials or algebraic expressions much more manageable.