Problem 42 - Elementary Algebra

Question

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ y-5+y^{3}=3 y^{2}+2 $$

Step-by-Step Solution

Answer

Answer: The given equation is a cubic equation.

Step 1: Write down the given equation

Write down the equation $$ y-5+y^{3}=3 y^{2}+2 $$.

Step 2: Identify the highest degree

Identify the highest power of the variable 'y'. The highest power of 'y' in the equation is $$y^{3}$$. So, the highest degree is 3.

Step 3: Classify the equation

Since the highest degree is 3, the equation is cubic.

Key Concepts

Cubic Equations

Cubic equations are polynomial equations that have a degree of three. This means that the highest exponent of the variable in the equation is three. The general form of a cubic equation is \( ax^3 + bx^2 + cx + d = 0 \), where \( a eq 0 \).

• The "cubic" term refers to the third degree, stemming from the Latin word 'cubus', which translates to cube. Imagine using cubes to model the equation visually, hence the term "cubic."
• Cubic equations often involve variables raised to the power of three, which can make them more complex than linear or quadratic equations.
• Solving cubic equations might require factoring, graphing, or using specific mathematical formulas like Cardano's method.

Understanding cubic equations is crucial in mathematics as they help model real-world scenarios involving volume calculations and physics problems.

Degree of an Equation

The degree of an equation is a vital aspect in understanding polynomial equations. It is determined by the highest power of the variable in the equation.

• In a polynomial equation with one variable, such as \( y - 5 + y^3 = 3y^2 + 2 \), to find the degree, look for the highest exponent of \( y \). Here, the degree is 3 since the term \( y^3 \) has the highest exponent.
• The degree reveals a lot about the equation's behavior. For example, a higher degree usually indicates more complex behavior in the graph of the equation.
• Only the term with the highest degree impacts the classification of the polynomial, not the constant terms or coefficients.

Degrees help in solving equations, graphing them, and predicting how changes in equations affect their solutions. Understanding the degree is crucial for determining the methods used for solving or simplifying the equations.

Classifying Equations by Degree

Classifying equations by degree helps in identifying and solving them more effectively. Here's how the classification works by degree.

• **Linear Equations**: These are equations with a degree of 1. They can be written in the form \( ax + b = 0 \) and produce straight lines when graphed.
• **Quadratic Equations**: These have a degree of 2. The standard form is \( ax^2 + bx + c = 0 \) and their graphs are parabolas.
• **Cubic Equations**: As we discussed, they have a degree of 3 and have a more complex graph that can have up to two turning points.

Classifying by degree is fundamental in mathematics as it guides the methods to solve them. Recognizing whether an equation is linear, quadratic, or cubic will influence calculations, expectations of solution types, and graphing properties. This classification streamlines the approach one uses to tackle different polynomial problems.