Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (2 x+3 y)^{3} $$

Step-by-Step Solution

Answer

The expression is \(8x^3 + 36x^2y + 54xy^2 + 27y^3\).

Step 1: Understand the Cubed Binomial Formula

To expand \((2x + 3y)^3\), we need to use the binomial theorem for cubes, which states: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Here, \(a = 2x\) and \(b = 3y\).

Step 2: Calculate Each Term

Using the binomial formula, calculate each term:- The first term is \((2x)^3 = 8x^3\).- The second term is \(3 imes (2x)^2 imes 3y = 36x^2y\).- The third term is \(3 imes 2x imes (3y)^2 = 54xy^2\).- The fourth term is \((3y)^3 = 27y^3\).

Step 3: Combine the Terms

Combine all the terms from Step 2 into a single polynomial. The expression thus becomes: \[8x^3 + 36x^2y + 54xy^2 + 27y^3\].

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra used to expand expressions that are raised to a power. Simply put, if you have an expression like \((a + b)^n\), the Binomial Theorem helps you express it as a sum of terms involving coefficients and powers of \(a\) and \(b\). Here is the basic form for any positive integer \(n\): \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]- **\(\binom{n}{k}\):** This notation represents the binomial coefficient, also known as combinations, which tells us how many ways to choose \(k\) elements from a set of \(n\) elements. It's calculated as:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]- **Term Structure:** Each term is made of a coefficient times \(a\) raised to some power times \(b\) raised to the complement power. The Binomial Theorem is vital because it simplifies the expansion of binomials without having to do tedious multiplication. This concept has broad applications in mathematics and science, helping to solve complex algebraic expressions efficiently.

Cubic Expansion

Cubic expansion is specifically about expanding expressions that are raised to the third power, such as \((a + b)^3\). This process uses principles from the Binomial Theorem to calculate the expanded form. The formula for cubic expansion is:\[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]Here's how the terms break down:

• \(a^3\): This represents the cube of the first term.
• \(3a^2b\): First, square the first term, then multiply it by the second term and by 3 for the coefficient. This accounts for the combinations or arrangements of these elements.
• \(3ab^2\): Multiply the first term by the square of the second term then by 3.
• \(b^3\): This is simply the cube of the second term.

Understanding cubic expansion helps to translate complex multiplication of binomial expressions into a sum of easier terms. This makes it easier to solve or simplify other higher-order polynomials and algebraic equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations. They can involve addition, subtraction, multiplication, division, and exponents. Expressions like \((2x + 3y)^3\) consist of terms; each term is a product of a number (coefficient) and variables raised to exponents.Breaking down an algebraic expression helps in solving or simplifying the equation. For example, the expression \((2x + 3y)^3\) becomes a polynomial when expanded:1. **Identify Terms:** Recognize the base terms, in this case, \(2x\) and \(3y\).2. **Apply Operations:** Use operations (like the binomial theorem) to expand the expression.3. **Simplify:** Combine like terms to simplify the expression and make calculations easier.Algebraic expressions are fundamental in mathematics, forming the basis for equations, functions, and models used in various scientific disciplines.