Tamara And Clyde Got Different Answers When Dividing $2x^4 + 7x^3 - 18x^2 + 11x - 2$ By $2x^2 - 3x + 1$. Analyze Their Individual Work.Tamara's Work:$\[ \begin{array}{l} 2x^2 - 3x + 1 \quad | \quad X^2 + 2x - 5.5

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Introduction

In the world of mathematics, division is a fundamental operation that helps us simplify complex expressions and solve equations. However, even with the right approach, different individuals may arrive at different answers. In this article, we will delve into the work of Tamara and Clyde, two students who got different results when dividing the polynomial $2x^4 + 7x^3 - 18x^2 + 11x - 2$ by $2x^2 - 3x + 1$. We will analyze their individual work, identify the mistakes, and provide a step-by-step solution to the problem.

Tamara's Work

Tamara's approach to dividing the polynomial was to use long division. She started by dividing the leading term of the dividend, $2x^4$, by the leading term of the divisor, $2x^2$. This gave her a quotient of $x^2$.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the dividend and divisor
dividend = 2*x**4 + 7*x**3 - 18*x**2 + 11*x - 2
divisor = 2*x**2 - 3*x + 1

# Perform long division
quotient = sp.div(dividend, divisor)

However, Tamara made a mistake in her calculation. She wrote the next term of the quotient as $2x$, but she forgot to multiply the entire divisor by this term and subtract it from the dividend.

# Tamara's incorrect calculation
tamara_quotient = x**2 + 2*x - 5.5

Discussion

Tamara's mistake was a common one. She failed to multiply the entire divisor by the next term of the quotient and subtract it from the dividend. This resulted in an incorrect quotient.

Clyde's Work

Clyde, on the other hand, used a different approach to dividing the polynomial. He started by factoring the divisor, $2x^2 - 3x + 1$, into $(2x - 1)(x - 1)$.

# Factor the divisor
factor1 = 2*x - 1
factor2 = x - 1

He then used the factored form of the divisor to rewrite the dividend as a product of the divisor and a quotient.

# Rewrite the dividend as a product of the divisor and a quotient
quotient = (2*x**2 + 3*x - 2) / (2*x - 1)

Clyde's approach was correct, but he made a mistake in his calculation. He wrote the quotient as $(2x^2 + 3x - 2) / (2x - 1)$, but he forgot to simplify the expression.

# Clyde's incorrect calculation
clyde_quotient = (2*x**2 + 3*x - 2) / (2*x - 1)

Conclusion

In conclusion, Tamara and Clyde's different answers were due to their individual mistakes. Tamara failed to multiply the entire divisor by the next term of the quotient and subtract it from the dividend, while Clyde forgot to simplify the expression.

Step-by-Step Solution

To solve the problem, we need to perform long division using the correct approach. We will divide the leading term of the dividend, $2x^4$, by the leading term of the divisor, $2x^2$. This gives us a quotient of $x^2$.

# Perform long division
quotient = sp.div(dividend, divisor)

We then multiply the entire divisor by the next term of the quotient, $2x$, and subtract it from the dividend.

# Multiply the entire divisor by the next term of the quotient
next_term = 2*x
product = next_term * divisor

# Subtract the product from the dividend
remainder = dividend - product

We repeat this process until we have divided the entire dividend.

# Repeat the process until we have divided the entire dividend
while remainder.degree() >= divisor.degree():
    next_term = remainder.coeff(x, divisor.degree() - 1)
    product = next_term * divisor
    remainder = remainder - product

The final quotient is the result of the division.

# The final quotient is the result of the division
final_quotient = quotient

Final Answer

The final answer is $\boxed{x^2 + 3x - 2}$.

Discussion Category

This problem falls under the category of algebra and polynomial division.

References

  • [1] Sympy: A Python library for symbolic mathematics.
  • [2] Khan Academy: Polynomial division.

Note

Introduction

In our previous article, we analyzed the work of Tamara and Clyde, two students who got different results when dividing the polynomial $2x^4 + 7x^3 - 18x^2 + 11x - 2$ by $2x^2 - 3x + 1$. We identified the mistakes in their individual work and provided a step-by-step solution to the problem. In this article, we will answer some frequently asked questions related to polynomial division.

Q&A Session

Q: What is polynomial division?

A: Polynomial division is a mathematical operation that involves dividing a polynomial by another polynomial. It is used to simplify complex expressions and solve equations.

Q: Why do we need to perform polynomial division?

A: Polynomial division is necessary when we need to simplify complex expressions or solve equations. It helps us to reduce the degree of the polynomial and make it easier to work with.

Q: What are the steps involved in polynomial division?

A: The steps involved in polynomial division are:

  1. Divide the leading term of the dividend by the leading term of the divisor.
  2. Multiply the entire divisor by the quotient and subtract it from the dividend.
  3. Repeat the process until we have divided the entire dividend.

Q: What is the difference between polynomial division and long division?

A: Polynomial division and long division are similar, but they are used for different types of numbers. Long division is used for integers, while polynomial division is used for polynomials.

Q: Can we use polynomial division to divide a polynomial by a non-polynomial expression?

A: No, we cannot use polynomial division to divide a polynomial by a non-polynomial expression. Polynomial division is only used for dividing polynomials by other polynomials.

Q: What are some common mistakes to avoid when performing polynomial division?

A: Some common mistakes to avoid when performing polynomial division include:

  • Forgetting to multiply the entire divisor by the quotient and subtract it from the dividend.
  • Not repeating the process until we have divided the entire dividend.
  • Not simplifying the expression after performing the division.

Q: How can we check if our answer is correct?

A: We can check if our answer is correct by multiplying the quotient by the divisor and adding the remainder. If the result is equal to the original dividend, then our answer is correct.

Q: What are some real-world applications of polynomial division?

A: Polynomial division has many real-world applications, including:

  • Simplifying complex expressions in physics and engineering.
  • Solving equations in mathematics and computer science.
  • Modeling real-world phenomena in economics and finance.

Conclusion

In conclusion, polynomial division is a powerful tool that helps us to simplify complex expressions and solve equations. By understanding the steps involved in polynomial division and avoiding common mistakes, we can ensure that our answers are correct and accurate.

References

  • [1] Sympy: A Python library for symbolic mathematics.
  • [2] Khan Academy: Polynomial division.
  • [3] Wolfram Alpha: Polynomial division.

Note

This article is for educational purposes only. The questions and answers are fictional and not intended to be used in real-world applications.