Problem 9 - Basic Technical Mathematics with Calculus

Question

Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify \(a, b,\) and \(c,\) with \(a>0 .\) Otherwise, explain why the resulting form is not quadratic. $$y^{2}(y-2)=3(y-2)$$

Step-by-Step Solution

Answer

The equation is not quadratic; it is cubic with the highest degree 3.

Step 1: Expand the Left Side

Start with the original equation: \[ y^2(y - 2) = 3(y - 2) \]Expand the left side by distributing \(y^2\):\[ y^2 imes y - y^2 imes 2 = y^3 - 2y^2 \]Now the equation is:\[ y^3 - 2y^2 = 3(y - 2) \]

Step 2: Expand the Right Side

Expand the right side of the equation:\[ 3(y - 2) = 3y - 6 \]Now the whole equation is:\[ y^3 - 2y^2 = 3y - 6 \]

Step 3: Rearrange Equation to Standard Form

Move all terms to one side of the equation to form a polynomial equation:\[ y^3 - 2y^2 - 3y + 6 = 0 \]

Step 4: Evaluate the Equation

The equation \( y^3 - 2y^2 - 3y + 6 = 0 \) is a cubic equation because the highest degree of variable \(y\) is 3. Quadratic equations have the form: \(ay^2 + by + c = 0\), where the highest degree is 2. Therefore, the given equation is not a quadratic.

Key Concepts

Polynomials

Polynomials are fundamental expressions in algebra, comprising variables and coefficients. They are formed by adding or subtracting terms, which are composed of a variable raised to a non-negative integer power and multiplied by a coefficient. In simpler terms, a polynomial is a sum of monomials, each having its own variable exponent and coefficient.

Here are some essential elements to remember about polynomials:

• **Terms:** Each individual part of a polynomial, such as \(5x^2\) or \(-3y\).
• **Coefficient:** The numerical factor multiplying the variable, like the "5" in \(5x^2\).
• **Exponent:** The power to which the variable is raised, such as the 2 in \(x^2\).
• **Constant term:** A term with no variable part, such as "+7" in \(x^2 + 2x + 7\).

Polynomials are named based on their degree, which is the value of the largest exponent. They are widely used in various fields, from solving equations to modeling real-world scenarios.

Cubic Equations

Cubic equations are a special type of polynomial equation where the highest power of the variable is three. This means that the equation has a degree of three. The standard form of a cubic equation is \(ax^3 + bx^2 + cx + d = 0\), where \(a, b, c,\) and \(d\) are constants, and \(a eq 0\).

• **Roots of a Cubic Equation:** A cubic equation can have up to three real roots because an equation of degree three is theoretically expected to have three solutions. These solutions could be all real numbers or include complex numbers.
• **Graph Characteristics:** The graph of a cubic function is generally a curve with at least one point of inflection, where the curvature changes direction. With three real roots, the graph will intersect the x-axis at those points.
• **Solving Methods:** To solve cubic equations, methods include factoring (if possible), using synthetic division, or applying techniques such as Cardano's formula or the use of graphing.

Cubic equations are important in various applications, including physics, engineering, and computer graphics, where they help model three-dimensional space.

Degree of a Polynomial

The **degree of a polynomial** is a key concept that describes the highest power of the variable in the polynomial expression. It is a crucial measure because it gives insights into the polynomial's behavior and properties.

Determining the degree of a polynomial is straightforward:

• **Identifying Degree:** Look for the term with the highest power on the variable. In the expression \(3x^4 + 5x^3 - 2x + 7\), the degree is 4.
• **Significance:** The degree tells us the maximum number of solutions a polynomial equation can have. For example, a third-degree polynomial (cubic) can have up to three roots.
• **Even vs. Odd Degrees:** Polynomials of even degree typically have graphs that behave similarly at both ends (both ends rise or fall together), while those of odd degree (like cubic equations) diverge at the ends (one end rises and the other falls).

Understanding the degree is crucial for solving polynomial equations and interpreting their graphs. It helps predict the possible number of roots and the general shape of the graph on a coordinate plane.