Problem 43 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The expanded expression is \(8x^9 - 12x^6y^2 + 6x^3y^4 - y^6\).

Step 1: Identify the Degree of the Expansion

The expression we need to expand is \((2x^3 - y^2)^3\). Notice that we are raising the binomial expression to the power 3.

Step 2: Identify the Row in Pascal's Triangle

Pascal's Triangle provides binomial coefficients. For an expansion with degree 3, we use the fourth row of Pascal's Triangle (since we start counting from 0), which is \([1, 3, 3, 1]\).

Step 3: Apply Binomial Theorem

The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). For this expression, let \(a = 2x^3\) and \(b = -y^2\).

Step 4: Calculate Each Term of the Expansion

- First term: \(\binom{3}{0} (2x^3)^3 (-y^2)^0 = 1 \cdot 8x^9 = 8x^9\).- Second term: \(\binom{3}{1} (2x^3)^2 (-y^2)^1 = 3 \cdot 4x^6 \cdot (-y^2) = -12x^6y^2\).- Third term: \(\binom{3}{2} (2x^3)^1 (-y^2)^2 = 3 \cdot 2x^3 \cdot y^4 = 6x^3y^4\).- Fourth term: \(\binom{3}{3} (2x^3)^0 (-y^2)^3 = 1 \cdot (-y^6) = -y^6\).

Step 5: Write the Expanded Expression

Combine the terms calculated: \(8x^9 - 12x^6y^2 + 6x^3y^4 - y^6\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions of the form \((a + b)^n\). It allows us to express such expansions as a sum of terms involving coefficients, powers of \(a\), and powers of \(b\). These coefficients can be obtained from Pascal's Triangle, which is a triangular array of numbers.
The numbers in Pascal’s Triangle are known as binomial coefficients. They represent the coefficients in the expansion, helping us easily determine the value of each term. The general formula for the expansion is:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Here, \( \binom{n}{k} \) represents the binomial coefficient, which is also the \(k\)-th entry of the \((n+1)\)-th row of Pascal's Triangle. Understanding this theorem makes it simple to handle complex binomial expansions.

Binomial Expansion

Binomial Expansion involves using the Binomial Theorem to express a binomial expression raised to a power in an expanded form. For example, if you have the expression \((a + b)^3\), you want to expand it into several terms. Each term in the expanded form is derived from the combination of:

• the binomial coefficient from Pascal's Triangle
• the two variables \(a\) and \(b\)
• the respective powers of \(a\) and \(b\).

In the given problem, \((2x^3 - y^2)^3\), we set \(a = 2x^3\) and \(b = -y^2\). By applying the coefficients from Pascal’s Triangle row number four \([1, 3, 3, 1]\), each term in the expansion of the expression is calculated sequentially by decreasing the power of \(a\) and increasing the power of \(b\). The methodical expansion results in:

• First term: \(8x^9\)
• Second term: \(-12x^6y^2\)
• Third term: \(6x^3y^4\)
• Fourth term: \(-y^6\)

Combining these, we achieve the expanded form: \(8x^9 - 12x^6y^2 + 6x^3y^4 - y^6\).

Polynomial Expression

A Polynomial Expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomials are fundamental in algebra and are instrumental for various calculations from simple to complex operations.
In our original problem, the expression \((2x^3 - y^2)^3\) represents a polynomial in two variables, \(x\) and \(y\). After expansion, it results in a polynomial with several terms. Each term in such a polynomial is typically a product of constants, powers of the variables along with the coefficients derived from the binomial coefficients.

• These polynomials can be manipulated in many different ways, such as addition, subtraction, and multiplication.
• Understanding the structure of polynomials, like in the expanded form \(8x^9 - 12x^6y^2 + 6x^3y^4 - y^6\), aids in interpreting, solving, and using them effectively in mathematical expressions and problems.

Understanding the polynomial's structure and how each term contributes to the full expression is critical for students studying mathematics. It equips them to handle more complex algebraic expressions efficiently.