2x^4 + 5x^3 + X - 1 Divided By X^2 -2x +1

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Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, and understanding how to perform it is essential for solving various mathematical problems. In this article, we will focus on dividing the polynomial 2x^4 + 5x^3 + x - 1 by x^2 -2x +1. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding the Polynomials

Before we begin the division process, let's take a closer look at the two polynomials involved:

  • Dividend: 2x^4 + 5x^3 + x - 1
  • Divisor: x^2 -2x +1

The dividend is the polynomial that we want to divide, and the divisor is the polynomial by which we want to divide it.

Step 1: Write the Dividend and Divisor in Standard Form

To begin the division process, we need to write both the dividend and divisor in standard form. The standard form of a polynomial is the form in which the terms are arranged in descending order of their exponents.

  • Dividend: 2x^4 + 5x^3 + x - 1
  • Divisor: x^2 -2x +1

Step 2: Divide the Leading Term of the Dividend by the Leading Term of the Divisor

The first step in the division process is to divide the leading term of the dividend by the leading term of the divisor. In this case, the leading term of the dividend is 2x^4, and the leading term of the divisor is x^2.

  • Leading term of dividend: 2x^4
  • Leading term of divisor: x^2

To divide 2x^4 by x^2, we need to multiply 2x^4 by the reciprocal of x^2, which is 1/x^2.

  • Result of division: 2x^2

Step 3: Multiply the Result by the Divisor and Subtract the Product from the Dividend

Next, we need to multiply the result of the division (2x^2) by the divisor (x^2 -2x +1) and subtract the product from the dividend.

  • Product of division result and divisor: 2x2(x2 -2x +1) = 2x^4 -4x^3 +2x^2
  • Subtraction: (2x^4 + 5x^3 + x - 1) - (2x^4 -4x^3 +2x^2) = 9x^3 -2x^2 + x - 1

Step 4: Repeat the Process with the Result of the Subtraction

We now repeat the process with the result of the subtraction (9x^3 -2x^2 + x - 1) as the new dividend.

  • New dividend: 9x^3 -2x^2 + x - 1
  • Divisor: x^2 -2x +1

We repeat the process by dividing the leading term of the new dividend (9x^3) by the leading term of the divisor (x^2).

  • Leading term of new dividend: 9x^3
  • Leading term of divisor: x^2

To divide 9x^3 by x^2, we need to multiply 9x^3 by the reciprocal of x^2, which is 1/x^2.

  • Result of division: 9x

Step 5: Multiply the Result by the Divisor and Subtract the Product from the New Dividend

Next, we need to multiply the result of the division (9x) by the divisor (x^2 -2x +1) and subtract the product from the new dividend.

  • Product of division result and divisor: 9x(x^2 -2x +1) = 9x^3 -18x^2 +9x
  • Subtraction: (9x^3 -2x^2 + x - 1) - (9x^3 -18x^2 +9x) = 16x^2 -8x - 1

Step 6: Repeat the Process with the Result of the Subtraction

We now repeat the process with the result of the subtraction (16x^2 -8x - 1) as the new dividend.

  • New dividend: 16x^2 -8x - 1
  • Divisor: x^2 -2x +1

We repeat the process by dividing the leading term of the new dividend (16x^2) by the leading term of the divisor (x^2).

  • Leading term of new dividend: 16x^2
  • Leading term of divisor: x^2

To divide 16x^2 by x^2, we need to multiply 16x^2 by the reciprocal of x^2, which is 1/x^2.

  • Result of division: 16

Step 7: Multiply the Result by the Divisor and Subtract the Product from the New Dividend

Next, we need to multiply the result of the division (16) by the divisor (x^2 -2x +1) and subtract the product from the new dividend.

  • Product of division result and divisor: 16(x^2 -2x +1) = 16x^2 -32x +16
  • Subtraction: (16x^2 -8x - 1) - (16x^2 -32x +16) = 24x -17

Conclusion

In this article, we have walked through the process of dividing the polynomial 2x^4 + 5x^3 + x - 1 by x^2 -2x +1. We have broken down the process into manageable steps and provided a clear explanation of each step. By following these steps, you should be able to divide polynomials with ease.

Final Answer

The final answer to the division problem is:

(2x^4 + 5x^3 + x - 1) / (x^2 -2x +1) = 2x^2 + 9x + 16 + (24x -17) / (x^2 -2x +1)

Q: What is polynomial division?

A: Polynomial division is a mathematical operation that involves dividing one polynomial by another. It is a crucial operation in algebra that helps us simplify complex expressions and solve equations.

Q: Why do we need to divide polynomials?

A: We need to divide polynomials to simplify complex expressions, solve equations, and find the roots of a polynomial. Polynomial division is a fundamental concept in algebra that helps us understand the properties of polynomials and their behavior.

Q: What are the steps involved in dividing polynomials?

A: The steps involved in dividing polynomials are:

  1. Write the dividend and divisor in standard form.
  2. Divide the leading term of the dividend by the leading term of the divisor.
  3. Multiply the result by the divisor and subtract the product from the dividend.
  4. Repeat the process with the result of the subtraction as the new dividend.
  5. Continue the process until the degree of the remainder is less than the degree of the divisor.

Q: What is the remainder in polynomial division?

A: The remainder in polynomial division is the result of the subtraction in the last step of the division process. It is a polynomial of degree less than the degree of the divisor.

Q: Can the remainder be divided further?

A: No, the remainder cannot be divided further by the divisor. The remainder is a polynomial of degree less than the degree of the divisor, and it cannot be divided further.

Q: What is the quotient in polynomial division?

A: The quotient in polynomial division is the result of the division process, excluding the remainder. It is a polynomial that represents the result of dividing the dividend by the divisor.

Q: How do we check the result of polynomial division?

A: To check the result of polynomial division, we need to multiply the quotient by the divisor and add the remainder. If the result is equal to the dividend, then the division is correct.

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

A: Some common mistakes to avoid in polynomial division are:

  • Not writing the dividend and divisor in standard form.
  • Not dividing the leading term of the dividend by the leading term of the divisor.
  • Not multiplying the result by the divisor and subtracting the product from the dividend.
  • Not repeating the process with the result of the subtraction as the new dividend.
  • Not checking the result of the division.

Q: How do we apply polynomial division in real-world problems?

A: Polynomial division is applied in various real-world problems, such as:

  • Simplifying complex expressions in physics and engineering.
  • Solving equations in economics and finance.
  • Finding the roots of a polynomial in computer science and data analysis.
  • Modeling population growth and decay in biology and ecology.

Q: What are some advanced topics in polynomial division?

A: Some advanced topics in polynomial division include:

  • Synthetic division.
  • Long division of polynomials.
  • Division of polynomials with complex coefficients.
  • Division of polynomials with rational coefficients.

Conclusion

In this article, we have answered some frequently asked questions about dividing polynomials. We have covered the basics of polynomial division, including the steps involved, the remainder, and the quotient. We have also discussed some common mistakes to avoid and how to apply polynomial division in real-world problems. Finally, we have touched on some advanced topics in polynomial division.