Problem 14 - Calculus

Question

Simplify the quantities using \(m(z)=z^{2}\). $$m(z+1)-m(z)$$

Step-by-Step Solution

Answer

The simplified expression is \(2z + 1\).

Step 1: Apply the function to the expression

We are given the function \(m(z) = z^2\) and asked to simplify \(m(z + 1) - m(z)\). Start by applying \(m\) to \(z+1\):\[ m(z+1) = (z+1)^2. \]

Step 2: Expand the squared term

Expand \((z+1)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\). Here \(a=z\) and \(b=1\):\[ (z+1)^2 = z^2 + 2z + 1. \]

Step 3: Substitute in Original Expression

Substitute \(m(z+1) = z^2 + 2z + 1\) and \(m(z) = z^2\) into the expression \(m(z+1) - m(z)\):\[ m(z+1) - m(z) = (z^2 + 2z + 1) - z^2. \]

Step 4: Simplify the Expression

Simplify the expression by canceling out the \(z^2\) terms:\[ (z^2 + 2z + 1) - z^2 = 2z + 1. \]

Key Concepts

Function Simplification

Function simplification is the process of making expressions more straightforward, removing unnecessary complexity while maintaining their original value. In mathematics, especially in algebra, simplification helps us better understand the behavior of functions. It makes it easier to identify patterns and perform calculations.

• In the exercise, we started with a complex expression: \( m(z+1) - m(z) \). This expression involves applying a function to different arguments and then finding the difference between the outcomes.
• The function given, \( m(z) = z^2 \), suggests we're dealing with squares. Instead of directly substituting and calculating, simplifying reveals underlying relationships.

Function simplification often involves manipulating algebraic terms, canceling like terms, and utilizing identities like \( (a + b)^2 = a^2 + 2ab + b^2 \). The ultimate goal is to express the function in its simplest, most digestible form.

Algebraic Expressions

Algebraic expressions are a foundational concept in mathematics, representing quantities with variables, constants, and operations like addition and multiplication. They are essentially math phrases that can be simplified, expanded, or evaluated. Consider the expression \( z^2 + 2z + 1 \), an algebraic expression representing the square of \( (z+1) \).

• Each part of an expression, such as \( z^2 \), represents a term. Terms can be either constant, a variable raised to a power, or a product of both.
• The expression is often transformed through processes like expansion, distributive property application, or simplification to make it more manageable.

In algebra, translating statements into algebraic expressions lets us generalize problems and find solutions efficiently. Expressing the problem in terms of \( z^2 \), for example, simplifies calculations across different scenarios involving squares.

Binomial Expansion

Binomial expansion deals with expanding expressions that involve the sum of two terms raised to a power, such as \( (z+1)^2 \). The expansion uses the binomial theorem, a powerful tool in algebra, especially useful in transforming expressions into easier-to-handle forms.

• The basic binomial formula \( (a + b)^2 = a^2 + 2ab + b^2 \) allows for quick expansion of the square terms. This identity reveals the structure of multiplication in terms of addition and exponentially distributed terms.
• In the exercise, applying the formula to \( (z+1)^2 \) unfolds the original expression into \( z^2 + 2z + 1 \), which simplifies understanding and further manipulation.

Binomial expansion is key in algebra as it makes seemingly complex operations straightforward through established formulas, enabling easier computation of higher-degree polynomial functions. It’s a crucial step in various algebraic processes, such as factoring, solving equations, and, as in this exercise, simplifying expressions.