Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\((2 x-y)^{3}\)

Step-by-Step Solution

Answer

So, \( (2x - y)^3 = 8x^3 - 12x^2*y + 6x*y^2 - y^3 \)

Step 1: Identify the Values

Identify the values of \(a\) and \(b\) from the given expression. Here, \(a = 2x\) and \(b = y\).

Step 2: Apply Cube of a Binomial Formula

Apply the formula for the cube of a binomial, \( (a-b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3\), substituting \(a = 2x\) and \(b = y\) to obtain \( (2x-y)^3 = (2x)^3 - 3*(2x)^2*y + 3*(2x)*y^2 - y^3 \).

Step 3: Simplify the Expression

After substituting the values and multiplying, simplify the expression to obtain the final outcome. This gives: \( (2x-y)^3 = 8x^3 - 12x^2*y + 6x*y^2 - y^3 \).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental to grasping the basics of algebra. An algebraic expression is a mathematical phrase that can include numbers, variables, and operations (like addition, subtraction, multiplication, and division). The expression does not have an equality sign, as opposed to an equation. In the context of our exercise, (2x - y)^3, we are dealing with a special type of algebraic expression known as a binomial. A binomial is an expression containing two terms separated by a plus or minus sign, like 2x - y in this case.
When we compute the cube of this binomial, we're essentially figuring out what we get when we multiply an algebraic binomial by itself three times. This action embodies one of the core operations in algebra, which is manipulating expressions to reveal simplified or expanded forms. The cube of this particular binomial also represents a specific instance of a polynomial expansion, which will be discussed later.

Binomial Theorem

The binomial theorem is a quick way to expand binomials raised to any power without having to multiply the binomial by itself over and over again. For the cube of a binomial, there's a specific formula derived from this theorem: (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. Here, a and b can be any numbers or algebraic expressions. This formula is incredibly useful as a shortcut to find the result of a cubed binomial.
In our example (2x - y)^3, a is 2x and b is y. By applying the binomial theorem, you eliminate the need for extensive multiplication, making it simpler and much quicker to come to a solution. Understand that the coefficients of the terms in the expanded form (1, 3, 3, 1) are actually the elements of the third row of Pascal's Triangle, which is closely related to the binomial theorem.

Polynomial Expansion

The term polynomial expansion refers to the process of expressing a polynomial that has been raised to a power as a sum of terms with different powers, each multiplied by a coefficient. As seen in our cube of a binomial, polynomial expansion can be simplified by using formulas and theorems such as the binomial theorem.
When we expand (2x - y)^3, we apply the binomial theorem to produce a polynomial with four terms. Each term of the polynomial expansion represents a possible combination of multiplying the elements 2x and -y in (2x - y)(2x - y)(2x - y).
This method illustrates why understanding polynomial expansion is crucial: it helps us to convert an expression that seems daunting due to its exponentiation into a more digestible and useful form. Recognizing the patterns in these expansions will also make you more efficient in handling algebraic expressions in various mathematical contexts.