Question

Use the Binomial Theorem to write the binomial expansion. \(\left(x^3-y^2\right)^4\)

Short answer

The binomial expansion for \((x^3 - y^2)^4\) is \(x^{12} - 4x^9*y^2 + 6x^6*y^4 - 4x^3*y^6 + y^8\).

Step-by-step solution

Identify the terms

First, identify the terms in the binomial. In this case, the terms for the binomial expression are \(x^3\) and \(-y^2\). The power that the binomial is raised to is 4.

Use the Binomial Theorem formula

The Binomial Theorem (for a binomial raised to the power of 4) states that\((a + b)^4 = a^4 + 4a^3*b + 6a^2*b^2 + 4ab^3 + b^4\). Replace \(a\) and \(b\) with the terms identified in Step 1. In the formula, \(a = x^3\) and \(b = -y^2\).

Calculate the binomial expansion

Substitute \(a\) and \(b\) into the formula:\((x^3 - y^2)^4 = (x^3)^4 + 4(x^3)^3*(-y^2) + 6(x^3)^2*(-y^2)^2 + 4(x^3)*(-y^2)^3 + (-y^2)^4\). Now perform the power operations on the quantities in the expression.

Simplify the expression

The expression simplifies to \(x^{12} - 4x^9*y^2 + 6x^6*y^4 - 4x^3*y^6 + y^8\). This is the binomial expansion for the given expression.

Key concepts

Binomial Expansion

The binomial expansion is a mathematical method used to expand expressions that are raised to a power. It takes a simple binomial, such as \((a + b)^n\), and expands it into a series of terms without parentheses. This allows us to express the polynomial in a more detailed form.
The Binomial Theorem provides a formula to accomplish this. It states that:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• The terms \(\binom{n}{k}\) are the binomial coefficients, which can be calculated using combinations, specifically \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• The sum \(k\) runs from 0 to \(n\), which means each term in the expansion is formed by raising \(a\) and \(b\) to different powers that add up to \(n\).

This process allows us to handle more complicated expressions by simplifying them into a series of smaller, manageable terms. Through binomial expansion, algebraic expressions become easier to work with, particularly for higher powers.

Algebraic Expressions

At the core, algebraic expressions are combinations of numbers, variables, and mathematical operations. In math, we often need to manipulate these expressions to solve problems or simplify calculations.
In the context of the binomial theorem, the expressions involved include terms like \(x^3\) and \(-y^2\) from the original formula \((x^3 - y^2)^4\). Each term can stand alone or combine with others to form a larger equation, representing a polynomial or another type of mathematical expression.
Algebraic expressions can take on varied forms, from simple linear equations to more complex polynomial expansions. They are fundamental tools in mathematics because they allow representation and manipulation of real-world problems in an abstract form. This abstraction helps in generalizing concepts and developing formulas that apply to a broad range of problems.
Efficient use of algebraic expressions often involves rewriting them using techniques such as factoring, expanding, and simplifying, as demonstrated through binomial expansion.

Polynomial Expansion

Polynomial expansion is a technique used to express a polynomial, formed through binomial expansion, as a sum of terms consisting of coefficients and variables raised to various powers.
When we apply the binomial theorem, the original compact binomial becomes a polynomial with many terms. For example, expanding \((x^3 - y^2)^4\) will transform it into a lengthy polynomial:

• \(x^{12} - 4x^{9}y^{2} + 6x^{6}y^{4} - 4x^3y^6 + y^8\)

Each term in the polynomial is a product of a coefficient (resulting from binomial coefficients), along with powers of the original terms \(a\) and \(b\) from the binomial formula.
The polynomial expansion offers an organized way to visibly see each component derived from the single binomial expression. This makes it easier for mathematicians and students alike to simplify, evaluate, and even graph complex expressions. Understanding how a polynomial comes from the original binomial can aid in solving many practical problems in science, engineering, and everyday calculations.