Problem 15 - Elementary Algebra

Question

Observe the equations and state the relationship being expressed. $$ w=4 z^{3}-21 $$

Step-by-Step Solution

Answer

Answer: The relationship between the variables w and z is that w is equal to four times the cube of z, minus 21.

Step 1: Given equation

We are given the following equation: $$w=4z^3-21$$

Step 2: Relationship between variables

The equation expresses the relationship between w and z as: w is equal to four times the cube of z, minus 21.

Key Concepts

Polynomial Function

Polynomials are algebraic expressions that consist of variables raised to whole number powers, and coefficients that multiply those variables. A polynomial function is a specific type of function that is formed by polynomials. In its simplest form, a polynomial can have:

• Constant coefficients, which are simply numbers.
• Variable terms, which involve variables like \(x\), \(y\), or any other letter from the alphabet.
• A degree, which is determined by the highest power of the variable in the polynomial.

In the given equation \(w = 4z^3 - 21\):

• The highest power of \(z\) is 3, which makes it a polynomial of degree 3.
• The term \(4z^3\) is called the leading term, and 4 is the leading coefficient.
• The number -21 is the constant term, a key part of the polynomial that affects its vertical shift.

Polynomial functions are widely used in mathematics to model a variety of real-world situations, thanks to their ability to represent complex relationships between variables. The versatility of polynomial functions makes them an important foundational concept in algebra.

Cubic Equation

A cubic equation is a type of polynomial equation that specifically includes a term raised to the third power. It's called 'cubic' because the highest exponent in the equation is three. This makes cubic equations more challenging than linear or quadratic equations but also more versatile in their applications.In the equation \(w = 4z^3 - 21\):

• The term \(4z^3\) means this is a cubic function.
• In cubic functions, you can have up to three possible real roots or solutions where the function's value is zero.
• They often have a characteristic S-shaped curve when plotted graphically, which differs from the U-shape of quadratic equations.

Cubic equations can describe a range of phenomena from calculating volumes to modeling physical systems. Their flexibility allows them to handle more complex relationships than quadratic equations, making them essential tools in engineering and sciences.

Variable Relationship

In mathematics, understanding the relationship between variables is crucial for interpreting how changes in one variable affect another. This relationship is often expressed in the form of an equation.In the equation \(w = 4z^3 - 21\):

• \(w\) is dependent on \(z\) because any change in \(z\) will directly affect the value of \(w\).
• The term \(4z^3\) indicates that \(w\) changes as the cube of \(z\), magnifying changes in \(z\) significantly more than linear or squared terms would.
• The constant term \(-21\) shifts the entire function downwards by 21 units on a graph, isolating the variable effect \(4z^3\) contributes to \(w\).

Such equations encapsulate all manner of real-world scenarios, where one quantity depends on various factors. Understanding these relationships helps in predicting outcomes and making informed decisions in fields such as economics, physics, and environmental science.