Problem 29 - College Algebra

Question

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ \left(x^{3}-\sqrt{y}\right)^{8} $$

Step-by-Step Solution

Answer

The first three terms are: \(x^{24} - 8x^{21}\sqrt{y} + 28x^{18}y\)."}

Step 1: Identify Binomial Theorem Formula

The Binomial Theorem is given by: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient. We will use this to expand \((x^3 - \sqrt{y})^8\).

Step 2: Find the General Term

The general term in the expansion is \(\binom{8}{k} (x^3)^{8-k} (-\sqrt{y})^k\). We will calculate the terms for \(k = 0, 1, 2\).

Step 3: Calculate the First Term (k = 0)

For \(k=0\), the term is \(\binom{8}{0} (x^3)^8 (-\sqrt{y})^0 = 1 \cdot x^{24} \cdot 1 = x^{24}\).

Step 4: Calculate the Second Term (k = 1)

For \(k=1\), the term is \(\binom{8}{1} (x^3)^7 (-\sqrt{y})^1 = 8 \cdot x^{21} \cdot (-\sqrt{y}) = -8x^{21}\sqrt{y}\).

Step 5: Calculate the Third Term (k = 2)

For \(k=2\), the term is \(\binom{8}{2} (x^3)^6 (-\sqrt{y})^2 = 28 \cdot x^{18} \cdot (y) = 28x^{18}y\).

Step 6: Combine the First Three Terms

The first three terms of the binomial expansion are \(x^{24} - 8x^{21}\sqrt{y} + 28x^{18}y\).

Key Concepts

Binomial Coefficient

The binomial coefficient is a crucial concept when it comes to expanding binomials using the Binomial Theorem. It is represented as \(\binom{n}{k}\), which is read as "n choose k."

These coefficients essentially count the number of ways to choose \(k\) items from \(n\) total, without considering the order of selection. Mathematically, it's calculated as:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

The exclamation mark "!" denotes the factorial, meaning \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).

In the context of our problem, we use these coefficients to determine the numerical factors for each term in the expansion of \((x^3 - \sqrt{y})^8\). For instance, when \(k=1\), the binomial coefficient is \(\binom{8}{1} = 8\), which serves as a multiplier for the term \(x^{21} \sqrt{y}\).

This coefficient can significantly influence the size and sign of the terms in the polynomial expansion, making it an essential tool in polynomial algebra.

Binomial Expansion

The binomial expansion involves transforming an expression of the form \((a + b)^n\) into a sum of terms using the Binomial Theorem. Each term in the expanded polynomial results from a particular choice for the exponent in each factor of the product.Let's break down the expansion process:

• Identify the expression \((a+b)^n\). In our exercise, it's \((x^3 - \sqrt{y})^8\).
• Use the formula \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), which allows you to break down the binomial into separate terms.
• Each term is calculated by multiplying the binomial coefficient with the respective powers of \(a\) and \(b\).

For example, the second term in our exercise is calculated as:

• \(\binom{8}{1} (x^3)^7 (-\sqrt{y})^1 = -8x^{21}\sqrt{y}\).

The binomial expansion converts complicated expressions into manageable polynomials, which are easier to work with in algebraic contexts, such as solving equations or simplifying expressions.

Polynomial Algebra

Polynomial algebra deals with expressions involving variables raised to various power increments, connected through addition, subtraction, and multiplication. When expanding binomials using the Binomial Theorem, we create a polynomial with several terms.

This polynomial expansion is then used to perform various algebraic operations.
Important concepts in polynomial algebra include:

• Degree: The highest power of the variable in the polynomial, like \(x^{24}\) in our expanded term, has a degree of 24.
• Term: Each separate mathematical expression within the polynomial, for example, \(-8x^{21}\sqrt{y}\).
• Coefficient: The numerical factor preceding each term, such as 28 in \(28x^{18}y\).

Knowing how to manipulate these elements effectively is crucial in solving problems and equations in higher mathematics.

The Binomial Theorem allows us to add structured, methodical expansion to the toolkit of polynomial algebra, making complex calculations straightforward. The beauty of polynomial algebra lies in its ability to represent real-world phenomena so they can be analyzed and understood through mathematical reasoning.