Quadratic Equation
When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations and their types.
Demand and Supply Function
The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to understand the consumer behavior. The evaluation of supply side factors is important for the consumers in order to understand that what kind of combination of goods or what kind of goods and services he or she should consume in order to maximize his utility and minimize the cost. Therefore, in microeconomics both of these concepts are extremely important in order to have an idea that what exactly is going on in the economy.
![### Mathematical Exercise: Expanding Quotients Using Fractions
In this exercise, we will learn how to expand the given quotient using fractions. The problem presented is:
\[ \frac{4x^3 - 6x^2 + 6x}{(x-1)^2 (x+1)} \]
The objective here is to expand the given fraction, making the polynomial in the numerator more manageable by expressing it as partial fractions. This process involves breaking down a complex fraction into a sum of simpler fractions that are easier to work with, especially for integration or solving equations.
Let's walk through the steps for expanding this quotient:
1. **Factorize the Denominator:**
The denominator \((x-1)^2 (x+1)\) is already in factored form. This makes the next steps of finding partial fractions straightforward.
2. **Set up Partial Fractions:**
We express the given fraction as a sum of partial fractions:
\[ \frac{4x^3 - 6x^2 + 6x}{(x-1)^2 (x+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+1} \]
3. **Determine the Coefficients (A, B, and C):**
To find the values of A, B, and C, we must clear the denominators by multiplying through by \((x-1)^2 (x+1)\) and then solve for A, B, and C by equating the corresponding coefficients on both sides of the equation.
The detailed steps to find these coefficients involve algebraic manipulation and solving systems of equations, which can be an excellent exercise to understand polynomial operations and partial fractions better.
#### Tips for Students:
- Ensure you are comfortable with polynomial long division and factorization.
- Practice solving simultaneous equations as this technique is pivotal in finding values for the coefficients.
- Always recheck your work by combining your partial fractions to see if you get back the original fraction.
Understanding this process aids in simplifying complex rational expressions and is a vital skill in various advanced mathematical fields, including calculus and engineering mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F840b63eb-0f35-40d8-ad49-813cc38d2dec%2Fbcc62179-8ae6-48db-bad4-06e663213ac8%2F4fsxajo.jpeg&w=3840&q=75)
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