Problem 19 - Algebra Form and Function

Question

In Exercises \(19-25, p(z)=4 z^{3}-z\). Find the given values and simplify if possible. $$ p(0) $$

Step-by-Step Solution

Answer

Answer: The value of the polynomial function p(z) when z = 0 is p(0) = 0.

Step 1: Identify the function and the value to substitute for z

We are given the function p(z) = 4z^3 - z and asked to evaluate it at z=0.

Step 2: Substitute the value into the function and simplify

Substitute the value z = 0 into the function: p(0) = 4(0)^3 - (0)

Step 3: Simplify the expression

Simplify the expression by performing the operations: p(0) = 4(0) - 0 p(0) = 0 - 0 p(0) = 0 The result of evaluating the function p(z) at z=0 is p(0) = 0.

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept where you determine the value of a function by substituting a specific value for the variable. In mathematics, a function assigns each input to a unique output. To evaluate, simply substitute the input value into the function's equation.
For example, to evaluate the function \( p(z) = 4z^3 - z \) at \( z = 0 \), you replace every \( z \) with 0. Perform the necessary calculations to determine the result. This process helps understand how functions behave with different inputs and is crucial across varied math applications.

Simplifying Expressions

Simplifying expressions involves removing unnecessary complexity to make them more digestible. The goal is to reduce expressions to their most straightforward form. This can involve combining like terms, eliminating redundant parts, or performing arithmetic operations.
In the case of our function \( p(0) = 4(0)^3 - 0 \), simplifying begins by recognizing that any number raised to a power and multiplied by zero remains zero. Here’s what happens:

• Calculate \( 4(0)^3 \), which is \( 4 \times 0 = 0 \).
• Subtracting 0 leaves the expression at 0.

Simplification is especially useful in validating function results and solving equations efficiently.

Polynomial Evaluation

Polynomial evaluation involves finding the output of a polynomial function at a specific point. This is done by plugging a number into the polynomial and carefully computing the result.
Polynomials are algebraic expressions composed of variables and coefficients, involving operations of addition, subtraction, and multiplication.
For example, evaluating \( p(z) = 4z^3 - z \) at \( z = 0 \) involves these steps:

• Identify the polynomial terms.
• Substitute 0 for every \( z \).
• Compute \( 4(0)^3 - 0 \) to get 0.

Evaluation helps predict with certainty the behavior of the polynomial for different variable inputs. It is a practical tool used extensively in calculus and algebra.