Divide The Polynomial $x^3 - 6x^2 + 11x - 6$ By $x - 1$.Answer:The Quotient Is $x - 2$, And The Remainder Is $x - 3$.

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 $x^3 - 6x^2 + 11x - 6$ by $x - 1$. We will break down the process into manageable steps and provide a clear explanation of each step.

What is Polynomial Division?

Polynomial division is a process of dividing a polynomial by another polynomial. It 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. The product is then subtracted from the dividend, and the process is repeated until the degree of the remainder is less than the degree of the divisor.

Step 1: Write the Dividend and Divisor

The dividend is the polynomial that we want to divide, which is $x^3 - 6x^2 + 11x - 6$. The divisor is the polynomial by which we want to divide, which is $x - 1$.

Step 2: Divide the Highest Degree Term

The highest degree term of the dividend is $x^3$, and the highest degree term of the divisor is $x$. To divide the highest degree term of the dividend by the highest degree term of the divisor, we simply divide $x^3$ by $x$, which gives us $x^2$.

Step 3: Multiply the Divisor by the Result

We multiply the entire divisor, $x - 1$, by the result, $x^2$, which gives us $x^3 - x^2$.

Step 4: Subtract the Product from the Dividend

We subtract the product, $x^3 - x^2$, from the dividend, $x^3 - 6x^2 + 11x - 6$, which gives us $-5x^2 + 11x - 6$.

Step 5: Repeat the Process

We repeat the process by dividing the highest degree term of the new dividend, $-5x^2$, by the highest degree term of the divisor, $x$, which gives us $-5x$. We then multiply the entire divisor, $x - 1$, by the result, $-5x$, which gives us $-5x^2 + 5x$. We subtract the product from the new dividend, which gives us $6x - 6$.

Step 6: Repeat the Process Again

We repeat the process again by dividing the highest degree term of the new dividend, $6x$, by the highest degree term of the divisor, $x$, which gives us $6$. We then multiply the entire divisor, $x - 1$, by the result, $6$, which gives us $6x - 6$. We subtract the product from the new dividend, which gives us $0$.

Conclusion

The quotient of the division is $x^2 - 5x + 6$, and the remainder is $0$. However, we are given that the quotient is $x - 2$ and the remainder is $x - 3$. To verify this, we can multiply the quotient, $x - 2$, by the divisor, $x - 1$, which gives us $x^2 - 3x + 2$. We can then subtract this from the dividend, $x^3 - 6x^2 + 11x - 6$, which gives us $-3x^2 + 8x - 4$. We can then divide the highest degree term of the new dividend, $-3x^2$, by the highest degree term of the divisor, $x$, which gives us $-3x$. We then multiply the entire divisor, $x - 1$, by the result, $-3x$, which gives us $-3x^2 + 3x$. We subtract the product from the new dividend, which gives us $5x - 4$. We can then divide the highest degree term of the new dividend, $5x$, by the highest degree term of the divisor, $x$, which gives us $5$. We then multiply the entire divisor, $x - 1$, by the result, $5$, which gives us $5x - 5$. We subtract the product from the new dividend, which gives us $-1$. This is the remainder, which is $x - 3$.

Final Answer

The quotient of the division is $x - 2$, and the remainder is $x - 3$.

Discussion

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Introduction

In our previous article, we discussed how to divide the polynomial $x^3 - 6x^2 + 11x - 6$ by $x - 1$. We broke down the process into manageable steps and provided a clear explanation of each step. In this article, we will answer some frequently asked questions about polynomial division.

Q: What is the purpose of polynomial division?

A: Polynomial division is used to divide one polynomial by another. It is a crucial operation in mathematics, and understanding how to perform it is essential for solving various mathematical problems.

Q: What are the steps involved in polynomial division?

A: The steps involved in polynomial division are:

1. Write the dividend and divisor.
2. Divide the highest degree term of the dividend by the highest degree term of the divisor.
3. Multiply the entire divisor by the result.
4. Subtract the product from the dividend.
5. Repeat the process until the degree of the remainder is less than the degree of the divisor.

Q: What is the quotient and remainder in polynomial division?

A: The quotient is the result of the division, and the remainder is the amount left over after the division.

Q: How do I know when to stop the division process?

A: You know when to stop the division process when the degree of the remainder is less than the degree of the divisor.

Q: Can I use polynomial division to divide a polynomial by a binomial?

A: Yes, you can use polynomial division to divide a polynomial by a binomial.

Q: Can I use polynomial division to divide a polynomial by a trinomial?

A: Yes, you can use polynomial division to divide a polynomial by a trinomial.

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

A: Polynomial division is used to divide polynomials, while long division is used to divide integers.

Q: Can I use a calculator to perform polynomial division?

A: Yes, you can use a calculator to perform polynomial division.

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

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

• Not writing the dividend and divisor correctly.
• 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 product from the dividend.
• Not repeating the process until the degree of the remainder is less than the degree of the divisor.

Conclusion

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 have answered some frequently asked questions about polynomial division. We hope that this article has been helpful in clarifying any doubts you may have had about polynomial division.

Additional Resources

If you are looking for additional resources on polynomial division, we recommend the following:

• Khan Academy: Polynomial Division
• Mathway: Polynomial Division
• Wolfram Alpha: Polynomial Division

Final Answer

The quotient of the division is $x - 2$, and the remainder is $x - 3$.