Divide $10x^4 - 14x^3 - 10x^2 + 6x - 10$ By $x^3 - 3x^2 + X - 2$. The Quotient Is $\square X + (-11)$. What Is The Remainder, Expressed As $\square X^2 + \square X + \square $?

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Polynomial division is a fundamental operation in algebra, and mastering it is crucial for simplifying complex expressions, solving equations, and understanding the behavior of polynomial functions. In this article, we will delve into the process of dividing a fourth-degree polynomial by a lower-degree polynomial, specifically focusing on the example of dividing 10x4−14x3−10x2+6x−1010x^4 - 14x^3 - 10x^2 + 6x - 10 by x3−3x2+x−2x^3 - 3x^2 + x - 2. We will break down the steps involved, explain the underlying principles, and provide clear examples to help you grasp this essential skill.

Understanding Polynomial Division

Polynomial division is analogous to long division with numbers. The goal is to find the quotient and the remainder when one polynomial (the dividend) is divided by another polynomial (the divisor). The quotient represents the result of the division, while the remainder is the polynomial left over after the division is complete. In our case, the dividend is 10x4−14x3−10x2+6x−1010x^4 - 14x^3 - 10x^2 + 6x - 10 and the divisor is x3−3x2+x−2x^3 - 3x^2 + x - 2.

Step-by-Step Division Process

To divide polynomials, we use a process similar to long division. Here's how it works:

  1. Set up the division: Write the dividend inside the division symbol and the divisor outside. Make sure both polynomials are written in descending order of exponents. Insert placeholders (terms with a coefficient of 0) for any missing terms. In our example, we have:
             _________________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
  1. Divide the leading terms: Divide the leading term of the dividend (10x410x^4) by the leading term of the divisor (x3x^3). This gives us 10x10x, which is the first term of the quotient. Write this term above the division symbol.
             10x______________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
  1. Multiply the quotient term by the divisor: Multiply the first term of the quotient (10x10x) by the entire divisor (x3−3x2+x−2x^3 - 3x^2 + x - 2). This gives us 10x4−30x3+10x2−20x10x^4 - 30x^3 + 10x^2 - 20x. Write this result below the dividend, aligning terms with the same exponents.
             10x______________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                    10x^4 - 30x^3 + 10x^2 - 20x
  1. Subtract: Subtract the result from the dividend. This means changing the signs of the terms in the expression we just wrote and adding.
             10x______________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                    -(10x^4 - 30x^3 + 10x^2 - 20x)
                    _________________________
                            16x^3 - 20x^2 + 26x - 10
  1. Bring down the next term: Bring down the next term from the dividend (-10) and write it next to the result of the subtraction.
             10x______________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                    -(10x^4 - 30x^3 + 10x^2 - 20x)
                    _________________________
                            16x^3 - 20x^2 + 26x - 10
  1. Repeat steps 2-5: Now, repeat the process using the new polynomial (16x3−20x2+26x−1016x^3 - 20x^2 + 26x - 10) as the new dividend. Divide the leading term (16x316x^3) by the leading term of the divisor (x3x^3), which gives us 16. Write +16 as the next term in the quotient.
             10x + 16_________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                    -(10x^4 - 30x^3 + 10x^2 - 20x)
                    _________________________
                            16x^3 - 20x^2 + 26x - 10
  1. Multiply and subtract: Multiply 16 by the divisor (x3−3x2+x−2x^3 - 3x^2 + x - 2) to get 16x3−48x2+16x−3216x^3 - 48x^2 + 16x - 32. Subtract this from the current dividend.
             10x + 16_________________
x^3 - 3x^2 + x - 2 | 10x^4 - 14x^3 - 10x^2 + 6x - 10
                    -(10x^4 - 30x^3 + 10x^2 - 20x)
                    _________________________
                            16x^3 - 20x^2 + 26x - 10
                    -(16x^3 - 48x^2 + 16x - 32)
                    _________________________
                             28x^2 + 10x + 22
  1. Determine the remainder: The resulting polynomial (28x2+10x+2228x^2 + 10x + 22) is the remainder. We stop the division process when the degree of the remainder is less than the degree of the divisor. In this case, the degree of the remainder (2) is less than the degree of the divisor (3), so we are done.

Results of the Division

From the division process, we find that:

  • Quotient: 10x+1610x + 16
  • Remainder: 28x2+10x+2228x^2 + 10x + 22

Expressing the Result

We can express the result of the division as:

10x4−14x3−10x2+6x−10=(x3−3x2+x−2)(10x+16)+(28x2+10x+22)10x^4 - 14x^3 - 10x^2 + 6x - 10 = (x^3 - 3x^2 + x - 2)(10x + 16) + (28x^2 + 10x + 22)

This equation shows that the dividend is equal to the divisor multiplied by the quotient, plus the remainder.

Solving the Specific Problem

Now, let's apply this process to the specific problem given:

Divide 10x4−14x3−10x2+6x−1010x^4 - 14x^3 - 10x^2 + 6x - 10 by x3−3x2+x−2x^3 - 3x^2 + x - 2.

We have already performed the division above and found that:

The quotient is 10x+1610x + 16. The remainder is 28x2+10x+2228x^2 + 10x + 22.

Therefore, the answer to the problem is:

The quotient is 10x+1610x + 16. The remainder is 28x2+10x+2228x^2 + 10x + 22.

Key Concepts and Considerations

  • Descending order: Always write polynomials in descending order of exponents before dividing.
  • Placeholders: Use placeholders (terms with a coefficient of 0) for any missing terms to maintain proper alignment during the division process.
  • Remainder degree: The division process stops when the degree of the remainder is less than the degree of the divisor.
  • Checking the result: To check your work, you can multiply the quotient by the divisor and add the remainder. The result should be the original dividend.

Applications of Polynomial Division

Polynomial division has various applications in mathematics and other fields, including:

  • Simplifying expressions: Dividing polynomials can help simplify complex expressions and make them easier to work with.
  • Solving equations: Polynomial division can be used to find the roots of polynomial equations.
  • Factoring polynomials: If the remainder is zero, then the divisor is a factor of the dividend.
  • Graphing polynomials: Polynomial division can help identify asymptotes and other key features of polynomial graphs.

Conclusion

Polynomial division is a powerful tool for manipulating and understanding polynomials. By mastering this process, you will gain a deeper understanding of algebraic concepts and be able to solve a wider range of problems. Remember to practice regularly and pay attention to the details of the process. With practice, you will become confident in your ability to divide polynomials of any degree.

In this article, we have provided a comprehensive guide to dividing a fourth-degree polynomial by a lower-degree polynomial. We have broken down the steps involved, explained the underlying principles, and provided clear examples to help you grasp this essential skill. We encourage you to practice these techniques and apply them to various problems to further enhance your understanding.

Key Takeaways:

  • Polynomial division is analogous to long division with numbers.
  • The goal is to find the quotient and the remainder.
  • The division process involves dividing, multiplying, subtracting, and bringing down terms.
  • The remainder degree must be less than the divisor degree.
  • Polynomial division has various applications in mathematics and other fields.