Problem 28 - Algebra 2

Question

Expand each binomial. $$ (2 x+3 y)^{3} $$

Step-by-Step Solution

Answer

\((2x+3y)^3 = 8x^3 + 36x^2y + 54xy^2 + 27y^3\)

Step 1: Identify the Binomial

The given binomial is \((2x+3y)^{3}\) and we're required to expand it.

Step 2: Apply the Binomial Theorem

To expand this binomial, we will use the binomial theorem. Here a is \(2x\), b is \(3y\), and n is 3.

Step 3: Calculate

Using the binomial theorem we get: \((2x+3y)^3 = (2x)^3 + 3C1 * (2x)^2 * 3y + 3C2 * 2x * (3y)^2 + (3y)^3\) We calculate this to get: \((2x+3y)^3 = 8x^3 + 3*4*x^2*3y + 3*2x*9y^2 + 27y^3\) which simplifies to: \((2x+3y)^3 = 8x^3 + 36x^2y + 54xy^2 + 27y^3\)

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful method for expanding expressions that are raised to a power, specifically those of the form \((a + b)^n\). It's a fundamental concept in algebra that enables us to systematically expand binomials without multiplying the expression repeatedly.

The theorem states that any binomial \((a + b)^n\) can be expanded as:

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

Here, \(C(n, k)\) are the binomial coefficients, representing the various combinations of powers of \(a\) and \(b\) in the expansion. The binomial coefficients can be found using Pascal's Triangle or through the formula \(C(n, k) = \frac{n!}{k! \, (n-k)!}\).

In our example, \((2x + 3y)^3\), this method allows us to efficiently expand the expression using known coefficients and powers.

Algebraic Expressions

Algebraic expressions are mathematical phrases that combine numbers, variables, and operations (like addition and multiplication). They are the building blocks of algebra and allow us to create and solve equations.

In the context of binomial expansion, understanding the structure of algebraic expressions is crucial. In our binomial \((2x + 3y)^3\), we identify "\(2x\)" and "\(3y\)" as terms. These terms are made up of:

• \(2x\): which is "2" (a coefficient) multiplied by "\(x\)" (a variable).
• \(3y\): which is "3" (a coefficient) multiplied by "\(y\)" (a variable).

They can be combined, manipulated, and expanded using algebraic rules. Understanding how coefficients and variables interact is essential for functions such as expanding binomials using the Binomial Theorem.

The result of expansion, such as "\(8x^3 + 36x^2y + 54xy^2 + 27y^3\)", is also an algebraic expression that shows each term clearly.

Exponents

Exponents are a shorthand way to represent repeated multiplication of a number by itself. In the binomial theorem, exponents apply both to whole expressions and individual components within an expression.

For example, let's take a look at the binomial \((2x + 3y)^3\). Here, the exponent "3" denotes that the expression \((2x + 3y)\) is multiplied by itself three times. The concept of exponents makes handling such expressions much more manageable.

While expanding the binomial, each term in the expansion will have exponents applied on the individual components like \(2x\) and \(3y\):

• \((2x)^3\) becomes \(8x^3\) by cubing both "2" and "x".
• \((3y)^3\) becomes \(27y^3\) by cubing both "3" and "y".

Each term in the expansion results from multiplying these components with the appropriate coefficients from the binomial expansion. Mastery of exponents is critical in algebra, as it simplifies expressions and calculations significantly.