Problem 36 - Algebra 2

Question

Expand each binomial. $$ (3 x+5 y)^{3} $$

Step-by-Step Solution

Answer

(3x+5y)^3 = 27x^3 + 270x^2*y + 900x*y^2 + 125y^3

Step 1: Identify the coefficients and variables

From the equation, we can see that \(a = 3x\), \(b = 5y\), and \(n = 3\).

Step 2: Apply the binomial theorem

Now, let's apply the theorem: \((a + b)^n = a^n + n*a^{n-1}*b + ((n*(n-1))/2!)*a^{n-2}*b^2 + ... b^n\). Substituting a, b and n values into the equation, we get \((3x+5y)^3 = (3x)^3 + 3*(3x)^{3-1}*(5y) + ((3*(3-1))/2!)*(3x)^{3-2}*(5y)^2 + (5y)^3\). Simplifying this equation we obtain keypoints \((3x+5y)^3 = 27x^3 + 270x^2*y + 900x*y^2 + 125y^3\).

Step 3: Simplify the equation

Now, we have the expanded binomial which is simplified to \((3x+5y)^3 = 27x^3 + 270x^2*y + 900x*y^2 + 125y^3\). This is the expanded form of the given binomial and our final answer.

Key Concepts

Binomial Theorem

The Binomial Theorem is a way of expanding expressions raised to a power. It allows us to break down expressions like \((a + b)^n\) into simpler parts. This theorem shows how to express such a binomial raised to any power as a sum of terms. Each term includes a coefficient known as a binomial coefficient.A binomial expression consists of two terms: \(a\) and \(b\). When we expand \((a + b)^n\), we apply the Binomial Theorem which is expressed as:

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

Here, \(\binom{n}{k}\) is the binomial coefficient, expressed mathematically as:

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

This term calculates the combinations of selecting \(k\) items from \(n\) items without considering the order. Each term in the expansion has a combination of \(a\) and \(b\) to different powers depending upon \(k\).Understanding this theorem fully can help simplify seemingly complex polynomial expansions.

Polynomials

Polynomials are expressions made of variables and coefficients combined using addition, subtraction, and multiplication. Such expressions may also include non-negative integer exponents. They take a form like \(ax^n + bx^{n-1} + \, ...\, + zx^0\), where each individual term consists of a coefficient multiplied by a variable raised to an exponent.Polynomials can have:

• Different degrees - the highest exponent present determines the degree. For example, in \(3x^3 + 2x^2 + x + 5\), the degree is 3.
• Various terms - each part separated by a plus or minus sign is a term. \(3x^2\) and \(-x\) are terms in \(3x^2 - x + 4\).

Polynomials are utilized in various fields like physics, engineering, and finance for modeling situations where there are relationships between different quantities that change together. By understanding polynomials, we can simplify and solve equations better.

Mathematical Expressions

Mathematical expressions are combinations of numbers, symbols, and operators including addition and subtraction. They represent a specific value and can include variables, which are symbols that stand in for unknown values.Expressions should not be confused with equations:

• An expression itself does not include an equality sign, whereas an equation sets one expression equal to another.

For instance, \(3x + 5\) is an expression, while \(3x + 5 = 20\) forms an equation. Types of expressions include:

• Algebraic expressions - contain numbers, variables, and operators, like \(2x + 3y\).
• Numeric expressions - contain only numbers and operators, such as \(4 + 7\).

Understanding these expressions plays a crucial role in problem-solving and allows for the manipulation and solving of equations using different mathematical rules and theorems such as the Binomial Theorem. They provide a way to describe mathematical ideas and relationships symbolically, facilitating communication and problem-solving in mathematics.