Problem 20 - Algebra Form and Function

Question

p(z)=4 z^{3}-z. Find the given values and simplify if possible. $$ p(\sqrt{5}) $$

Step-by-Step Solution

Answer

Answer: The value of the given function when $$z = \sqrt{5}$$ is $$p(\sqrt{5}) = 19\sqrt{5}$$.

Step 1: Identify the given function

We are given the function: $$ p(z) = 4z^3 - z $$

Step 2: Substitute z with the square root of 5

Now we substitute z with the square root of 5 in the given function: $$ p(\sqrt{5}) = 4(\sqrt{5})^3 - (\sqrt{5}) $$

Step 3: Simplify the expression

Begin by simplifying the exponent in the expression: $$ (\sqrt{5})^3 = (\sqrt{5})(\sqrt{5})(\sqrt{5}) = 5\sqrt{5} $$ Then, substitute the simplified exponent back into the expression and further simplify: $$ p(\sqrt{5}) = 4(5\sqrt{5}) - (\sqrt{5}) = 20\sqrt{5} - \sqrt{5} $$ Finally, factor the expression to find the solution: $$ p(\sqrt{5}) = \sqrt{5}(20 - 1) = 19\sqrt{5} $$

Step 4: Write the final answer

The value of the given function for z equal to the square root of 5 is: $$ p(\sqrt{5}) = 19\sqrt{5} $$

Key Concepts

Function Evaluation

Function evaluation is a foundational concept in algebra and calculus. It involves taking a function, which is a rule or relation defined by a mathematical expression, and determining its value for particular inputs.
For the function given in our exercise, \( p(z) = 4z^3 - z \), 'evaluating' means calculating \( p(z) \) for specific values of \( z \). Key aspects of function evaluation include:

• Substitution: Replace the variable in the function's expression with the given number or expression. For example, in the exercise, we substitute \( z \) with \( \sqrt{5} \) to find \( p(\sqrt{5}) \).
• Following Operations: Once substituted, ensure to perform all necessary arithmetic operations present in the function according to the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
• Finding the Result: Arrive at a single value which expresses the function's output for the given input, like \( 19\sqrt{5} \) for \( p(\sqrt{5}) \).

Evaluating functions will often lead into the next concepts such as simplification or algebraic manipulation, which are crucial for reaching an understandable and simplified final expression.

Simplification of Expressions

Simplification is all about making expressions easier to work with and understand. An expression is simplified by making it as concise as possible without changing its value. Let's break it down using our example:In our exercise, after substituting \( z = \sqrt{5} \), we have:

• The expression \( p(\sqrt{5}) = 4(\sqrt{5})^3 - (\sqrt{5}) \)

Simplification then involves:

• Handling Exponents: The term \( (\sqrt{5})^3 \) means \( (\sqrt{5})(\sqrt{5})(\sqrt{5}) \), which can be simplified to \( 5\sqrt{5} \). This step is crucial as it transforms a complex expression into something more manageable.
• Arithmetic Operations: Substitute back the simplified form \( 5\sqrt{5} \) into the expression to get \( 4(5\sqrt{5}) - \sqrt{5} \), which can then be simplified to \( 20\sqrt{5} - \sqrt{5} \).
• Combining Like Terms: We notice that both terms in \( 20\sqrt{5} - \sqrt{5} \) involve \( \sqrt{5} \), allowing further simplification as \( \sqrt{5}(20 - 1) = 19\sqrt{5} \).

Through these steps, simplification helps express mathematical ideas cleanly and opens pathways to solving more complex problems efficiently.

Algebraic Manipulation

Algebraic manipulation involves rearranging and reworking expressions using algebraic rules and operations. This skill is essential for solving equations and deriving expressions in simpler forms or in forms that reveal new insights. Using our problem as a reference, we experience algebraic manipulation in several ways:

• Substitution of Expressions: Initiate manipulation by substituting complex parts of the equation. In our scenario, substituting \( z = \sqrt{5} \) into the function \( p(z) = 4z^3 - z \).
• Use of Exponent Rules: Apply rules like \((a^m)(a^n) = a^{m+n}\) to simplify powers, such as turning \( (\sqrt{5})^3 \) into \( 5\sqrt{5} \).
• Factorization: This is used to identify common factors in terms. For instance, recognizing \( \sqrt{5} \) is a common factor in both \( 20\sqrt{5} \) and \( \sqrt{5} \) allows it to be factored out, simplifying the expression to \( \sqrt{5}(20 - 1) = 19\sqrt{5} \).

Algebraic manipulation often involves simplifying expressions to a form that reveals more about their properties or the result. By understanding and practicing these manipulations, students can tackle both simple and complex mathematical problems with greater confidence and skill.