Divide \[$ X^3 - 3x^2 + X - 2 \$\] By \[$ 10x^4 - 14x^3 - 10x^2 + 6x - 10 \$\].The Quotient Is \[$\square\$\] \[$x+\$\] \[$\square\$\].The Remainder Is \[$\square\$\] \[$x^2+\$\]

Feb 27, 2025 by ADMIN 179 views

$Divide ${$ x^3 - 3x^2 + x - 2 \$}$ by ${$ 10x^4 - 14x^3 - 10x^2 + 6x - 10 \$}$$

Introduction

In this article, we will be performing polynomial division, which is a method used to divide one polynomial by another. This process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting that from the dividend. We will be dividing the polynomial ${$ x^3 - 3x^2 + x - 2 $}$ by ${$ 10x^4 - 14x^3 - 10x^2 + 6x - 10 $}$.

Step 1: Divide the Highest Degree Term

To begin the division process, we need to divide the highest degree term of the dividend by the highest degree term of the divisor. In this case, the highest degree term of the dividend is ${$ x^3 $}$ and the highest degree term of the divisor is ${$ 10x^4 $}$. We can divide ${$ x^3 $}$ by ${$ 10x^4 $}$ to get ${$ \frac{x^3}{10x4} = \frac{1}{10x} $}$.

Step 2: Multiply the Divisor by the Result

Next, we need to multiply the entire divisor by the result we obtained in the previous step. In this case, we need to multiply ${$ 10x^4 - 14x^3 - 10x^2 + 6x - 10 $}$ by ${$ \frac{1}{10x} $}$. This will give us ${$ x^3 - \frac{14}{10}x^2 - \frac{10}{10}x^2 + \frac{6}{10}x - \frac{10}{10} $}$, which simplifies to ${$ x^3 - \frac{24}{10}x^2 + \frac{6}{10}x - 1 $}$.

Step 3: Subtract the Result from the Dividend

Now, we need to subtract the result we obtained in the previous step from the dividend. In this case, we need to subtract ${$ x^3 - \frac{24}{10}x^2 + \frac{6}{10}x - 1 $}$ from ${$ x^3 - 3x^2 + x - 2 $}$. This will give us ${$ -\frac{24}{10}x^2 + \frac{6}{10}x - 1 - 3x^2 + x - 2 $}$, which simplifies to ${$ -\frac{54}{10}x^2 + \frac{7}{10}x - 3 $}$.

Step 4: Repeat the Process

We need to repeat the process of dividing the highest degree term of the result by the highest degree term of the divisor, multiplying the divisor by the result, and subtracting the result from the dividend. In this case, we need to divide ${$ -\frac{54}{10}x^2 $}$ by ${$ 10x^4 $}$, which gives us ${$ \frac{-54}{10}x^2 \div 10x^4 = \frac{-54}{10}x^2 \times \frac{1}{10x^4} = \frac{-54}{100x^2} $}$.

Step 5: Multiply the Divisor by the Result

Next, we need to multiply the entire divisor by the result we obtained in the previous step. In this case, we need to multiply ${$ 10x^4 - 14x^3 - 10x^2 + 6x - 10 $}$ by ${$ \frac{-54}{100x^2} $}$. This will give us ${$ -\frac{54}{10}x^6 + \frac{54}{10}x^5 + \frac{54}{10}x^4 - \frac{54}{10}x^3 - \frac{54}{10}x^2 + \frac{54}{10}x $}$.

Step 6: Subtract the Result from the Dividend

Now, we need to subtract the result we obtained in the previous step from the dividend. In this case, we need to subtract ${$ -\frac{54}{10}x^6 + \frac{54}{10}x^5 + \frac{54}{10}x^4 - \frac{54}{10}x^3 - \frac{54}{10}x^2 + \frac{54}{10}x $}$ from ${$ -\frac{54}{10}x^2 + \frac{7}{10}x - 3 $}$. This will give us ${$ -\frac{54}{10}x^2 + \frac{7}{10}x - 3 + \frac{54}{10}x^6 - \frac{54}{10}x^5 - \frac{54}{10}x^4 + \frac{54}{10}x^3 + \frac{54}{10}x^2 - \frac{54}{10}x $}$, which simplifies to ${$ \frac{54}{10}x^6 - \frac{54}{10}x^5 - \frac{54}{10}x^4 + \frac{54}{10}x^3 + \frac{6}{10}x - 3 $}$.

Step 7: Repeat the Process

We need to repeat the process of dividing the highest degree term of the result by the highest degree term of the divisor, multiplying the divisor by the result, and subtracting the result from the dividend. In this case, we need to divide ${$ \frac{54}{10}x^6 $}$ by ${$ 10x^4 $}$, which gives us ${$ \frac{54}{10}x^6 \div 10x^4 = \frac{54}{10}x^6 \times \frac{1}{10x^4} = \frac{54}{100}x^2 $}$.

Step 8: Multiply the Divisor by the Result

Next, we need to multiply the entire divisor by the result we obtained in the previous step. In this case, we need to multiply ${$ 10x^4 - 14x^3 - 10x^2 + 6x - 10 $}$ by ${$ \frac{54}{100}x^2 $}$. This will give us ${$ \frac{54}{10}x^8 - \frac{54}{10}x^7 - \frac{54}{10}x^6 + \frac{54}{10}x^5 + \frac{54}{10}x^4 - \frac{54}{10}x^2 $}$.

Step 9: Subtract the Result from the Dividend

Now, we need to subtract the result we obtained in the previous step from the dividend. In this case, we need to subtract ${$ \frac{54}{10}x^8 - \frac{54}{10}x^7 - \frac{54}{10}x^6 + \frac{54}{10}x^5 + \frac{54}{10}x^4 - \frac{54}{10}x^2 $}$ from ${$ \frac{54}{10}x^6 - \frac{54}{10}x^5 - \frac{54}{10}x^4 + \frac{54}{10}x^3 + \frac{6}{10}x - 3 $}$. This will give us ${$ \frac{54}{10}x^6 - \frac{54}{10}x^5 - \frac{54}{10}x^4 + \frac{54}{10}x^3 + \frac{6}{10}x - 3 - \frac{54}{10}x^8 + \frac{54}{10}x^7 + \frac{54}{10}x^6 - \frac{54}{10}x^5 - \frac{54}{10}x^4 $}$, which simplifies to ${$ \frac{54}{10}x^7 + \frac{6}{10}x - 3 $}$.

Step 10: Repeat the Process

We need to repeat the process of dividing the highest degree term of the result by the highest degree term of the divisor, multiplying the divisor by the result, and subtracting the result from the dividend. In this case, we need to divide ${$ \frac{54}{10}x^7 $}$ by ${$ 10x^4 $}$, which gives us ${$ \frac{54}{10}x^7 \div 10x^4 = \frac{54}{10}x^7 \times \frac{1}{10x^4} = \frac{54}{100}x^3 $}$.

Step 11: Multiply the Divisor by the Result

Next, we need to multiply the entire divisor by the result we obtained in the previous step. In this case, we need to multiply ${$ 10x^4 - 14x^3 - 10x^2

Q&A

Q: What is polynomial division?

A: Polynomial division is a method used to divide one polynomial by another. This process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting that from the dividend.

Q: What is the quotient of the division of ${$ x^3 - 3x^2 + x - 2 $}$ by ${$ 10x^4 - 14x^3 - 10x^2 + 6x - 10 $}$?

A: The quotient is ${$ x + \square $}$ ${$ \square $}$.

Q: What is the remainder of the division of ${$ x^3 - 3x^2 + x - 2 $}$ by ${$ 10x^4 - 14x^3 - 10x^2 + 6x - 10 $}$?

A: The remainder is ${$ \square $}$ ${$ x^2 + \square $}$.

Q: How do I perform polynomial division?

A: To perform polynomial division, you need to divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the entire divisor by the result, and subtract that from the dividend. You need to repeat this process until the degree of the remainder is less than the degree of the divisor.

Q: What are the steps involved in polynomial division?

A: The steps involved in polynomial division are:

Divide the highest degree term of the dividend by the highest degree term of the divisor.
Multiply the entire divisor by the result.
Subtract the result from the dividend.
Repeat steps 1-3 until the degree of the remainder is less than the degree of the divisor.

Q: What is the importance of polynomial division?

A: Polynomial division is an important concept in algebra and is used to simplify complex expressions and solve equations. It is also used in various fields such as engineering, physics, and computer science.

Q: Can polynomial division be used to divide polynomials with different degrees?

A: Yes, polynomial division can be used to divide polynomials with different degrees. However, the degree of the remainder will be less than the degree of the divisor.

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

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

Not dividing the highest degree term of the dividend by the highest degree term of the divisor.
Not multiplying the entire divisor by the result.
Not subtracting the result from the dividend.
Not repeating the process until the degree of the remainder is less than the degree of the divisor.

Q: How do I check my work when performing polynomial division?

A: To check your work when performing polynomial division, you can multiply the quotient by the divisor and add the remainder. If the result is equal to the dividend, then your work is correct.

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

A: Some real-world applications of polynomial division include:

Simplifying complex expressions in engineering and physics.
Solving equations in computer science.
Modeling population growth and decay in biology.
Analyzing data in statistics.

Q: Can polynomial division be used to divide polynomials with complex coefficients?

A: Yes, polynomial division can be used to divide polynomials with complex coefficients. However, the process is more complex and requires the use of complex numbers.

Q: What are some tips for performing polynomial division?

A: Some tips for performing polynomial division include:

Use a systematic approach to divide the highest degree term of the dividend by the highest degree term of the divisor.
Multiply the entire divisor by the result and subtract that from the dividend.
Repeat the process until the degree of the remainder is less than the degree of the divisor.
Check your work by multiplying the quotient by the divisor and adding the remainder.