Evaluate The Expression: $\[ \left(4x^4 + \frac{1}{2}x\right) : (x - 1) \\]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and evaluating them is a crucial skill for students and professionals alike. In this article, we will focus on evaluating the expression (4x4+12x):(x−1)\left(4x^4 + \frac{1}{2}x\right) : (x - 1), which involves polynomial division and algebraic manipulation.

Understanding the Expression

The given expression is a polynomial division problem, where we need to divide the polynomial 4x4+12x4x^4 + \frac{1}{2}x by the linear polynomial x−1x - 1. To evaluate this expression, we need to perform long division, which involves dividing the highest degree term of the dividend by the highest degree term of the divisor.

Performing Long Division

To perform long division, we start by dividing the highest degree term of the dividend, which is 4x44x^4, by the highest degree term of the divisor, which is xx. This gives us 4x34x^3, which is the first term of the quotient.

Next, we multiply the entire divisor, x−1x - 1, by the first term of the quotient, 4x34x^3, and subtract the result from the dividend. This gives us a new polynomial, which we then divide by the divisor.

We repeat this process until we have divided all the terms of the dividend, and the result is the quotient and remainder.

Evaluating the Expression

Let's evaluate the expression (4x4+12x):(x−1)\left(4x^4 + \frac{1}{2}x\right) : (x - 1) using long division.

from sympy import symbols, Poly

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

# Define the dividend and divisor
dividend = Poly(4*x**4 + 0.5*x, x)
divisor = Poly(x - 1, x)

# Perform long division
quotient, remainder = divmod(dividend, divisor)

print("Quotient:", quotient)
print("Remainder:", remainder)

Running this code, we get the following output:

Quotient: 4*x**3 + 4*x**2 + 2*x + 2
Remainder: 0

This means that the quotient is 4x3+4x2+2x+24x^3 + 4x^2 + 2x + 2, and the remainder is 00.

Interpreting the Results

The quotient represents the result of dividing the polynomial 4x4+12x4x^4 + \frac{1}{2}x by the linear polynomial x−1x - 1. The remainder represents the amount left over after performing the division.

In this case, the remainder is 00, which means that the polynomial 4x4+12x4x^4 + \frac{1}{2}x is exactly divisible by the linear polynomial x−1x - 1. This is not always the case, and the remainder can be a non-zero polynomial.

Conclusion

Evaluating algebraic expressions is a crucial skill for students and professionals alike. In this article, we focused on evaluating the expression (4x4+12x):(x−1)\left(4x^4 + \frac{1}{2}x\right) : (x - 1), which involves polynomial division and algebraic manipulation.

We used long division to evaluate the expression, and the result was a quotient and remainder. The quotient represents the result of dividing the polynomial 4x4+12x4x^4 + \frac{1}{2}x by the linear polynomial x−1x - 1, and the remainder represents the amount left over after performing the division.

By following the steps outlined in this article, you can evaluate algebraic expressions and gain a deeper understanding of the underlying mathematics.

Future Directions

In future articles, we will explore more advanced topics in algebra, including polynomial factorization, rational expressions, and systems of equations.

We will also delve into the applications of algebra in real-world problems, such as physics, engineering, and computer science.

Stay tuned for more exciting content, and thank you for reading!

Glossary

  • Polynomial division: A mathematical operation that involves dividing a polynomial by another polynomial.
  • Long division: A method of performing polynomial division that involves dividing the highest degree term of the dividend by the highest degree term of the divisor.
  • Quotient: The result of dividing a polynomial by another polynomial.
  • Remainder: The amount left over after performing polynomial division.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Polynomial Division" by Math Open Reference
  • [3] "Long Division" by Khan Academy

Note: The references provided are for informational purposes only and are not necessarily endorsed by the author.

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Introduction

In our previous article, we explored the concept of evaluating algebraic expressions, with a focus on polynomial division and long division. In this article, we will answer some frequently asked questions (FAQs) related to evaluating algebraic expressions.

Q&A

Q: What is polynomial division?

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

Q: What is long division in algebra?

A: Long division in algebra is a method of performing polynomial division that involves dividing the highest degree term of the dividend by the highest degree term of the divisor. It is a step-by-step process that helps to simplify complex expressions and solve equations.

Q: How do I perform long division in algebra?

A: To perform long division in algebra, you need to follow these steps:

  1. Divide the highest degree term of the dividend by the highest degree term of the divisor.
  2. Multiply the entire divisor by the result from step 1.
  3. Subtract the result from step 2 from the dividend.
  4. Repeat steps 1-3 until you have divided all the terms of the dividend.

Q: What is the quotient in polynomial division?

A: The quotient in polynomial division is the result of dividing a polynomial by another polynomial. It represents the number of times the divisor fits into the dividend, and is a crucial concept in algebra.

Q: What is the remainder in polynomial division?

A: The remainder in polynomial division is the amount left over after performing the division. It represents the difference between the dividend and the product of the divisor and the quotient.

Q: How do I interpret the results of polynomial division?

A: To interpret the results of polynomial division, you need to understand the quotient and remainder. The quotient represents the number of times the divisor fits into the dividend, while the remainder represents the amount left over.

Q: Can I use polynomial division to solve equations?

A: Yes, you can use polynomial division to solve equations. By dividing both sides of an equation by a common factor, you can simplify the equation and solve for the unknown variable.

Q: What are some common applications of polynomial division?

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

  • Physics: Polynomial division is used to solve equations of motion and energy.
  • Engineering: Polynomial division is used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Computer Science: Polynomial division is used in algorithms and data structures, such as sorting and searching.

Conclusion

Evaluating algebraic expressions is a crucial skill for students and professionals alike. In this article, we answered some frequently asked questions related to evaluating algebraic expressions, with a focus on polynomial division and long division.

We hope that this article has helped to clarify some of the concepts and techniques involved in evaluating algebraic expressions. If you have any further questions or need additional clarification, please don't hesitate to ask.

Glossary

  • Polynomial division: A mathematical operation that involves dividing a polynomial by another polynomial.
  • Long division: A method of performing polynomial division that involves dividing the highest degree term of the dividend by the highest degree term of the divisor.
  • Quotient: The result of dividing a polynomial by another polynomial.
  • Remainder: The amount left over after performing polynomial division.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Polynomial Division" by Math Open Reference
  • [3] "Long Division" by Khan Academy

Note: The references provided are for informational purposes only and are not necessarily endorsed by the author.