Use Synthetic Division To Divide:$ \left(-x^3 + 69x - 310\right) \div (x + 10) }$ Given Result ${ -x^2 + 79 - \frac{1100 X+10} }$Note { X \neq -10 $ $

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a powerful tool for simplifying complex polynomial expressions and is widely used in mathematics and engineering. In this article, we will use synthetic division to divide the polynomial $-x^3 + 69x - 310$ by $x + 10$ and obtain the result $-x^2 + 79 - \frac{1100}{x+10}$.

What is Synthetic Division?

Synthetic division is a method of dividing polynomials by linear factors. It is a shortcut method that eliminates the need for long division and is often used when dividing polynomials by factors of the form $(x - a)$ or $(x + a)$. The method involves using a single row of numbers to perform the division, rather than the multiple rows required for long division.

How to Perform Synthetic Division

To perform synthetic division, we need to follow these steps:

1. Write down the coefficients of the polynomial in a row, with the constant term on the right-hand side.
2. Write down the value of the linear factor in the middle of the row.
3. Multiply the value in the middle of the row by the first coefficient and write the result below the row.
4. Add the result from step 3 to the second coefficient and write the result below the row.
5. Repeat steps 3 and 4 until we reach the last coefficient.
6. The final result is the coefficients of the quotient polynomial.

Example: Dividing $-x^3 + 69x - 310$ by $x + 10$

Let's use synthetic division to divide the polynomial $-x^3 + 69x - 310$ by $x + 10$. We will follow the steps outlined above to perform the division.

Step 1: Write down the coefficients of the polynomial

The coefficients of the polynomial $-x^3 + 69x - 310$ are $-1$, $0$, $69$, and $-310$. We will write these coefficients in a row, with the constant term on the right-hand side.

-1 | 0 69 -310

Step 2: Write down the value of the linear factor

The value of the linear factor $x + 10$ is $-10$. We will write this value in the middle of the row.

-1 | 0 69 -310
-10

Step 3: Multiply the value in the middle of the row by the first coefficient

We will multiply the value in the middle of the row, $-10$, by the first coefficient, $-1$, and write the result below the row.

-1 | 0 69 -310
-10
  10

Step 4: Add the result from step 3 to the second coefficient

We will add the result from step 3, $10$, to the second coefficient, $0$, and write the result below the row.

-1 | 0 69 -310
-10
  10
  69

Step 5: Repeat steps 3 and 4 until we reach the last coefficient

We will repeat steps 3 and 4 until we reach the last coefficient.

-1 | 0 69 -310
-10
  10
  69
 -310
  310
 -79

Step 6: The final result is the coefficients of the quotient polynomial

The final result is the coefficients of the quotient polynomial. We will write these coefficients in a row, with the constant term on the right-hand side.

-1 | -79 0 0

Conclusion

In this article, we used synthetic division to divide the polynomial $-x^3 + 69x - 310$ by $x + 10$ and obtained the result $-x^2 + 79 - \frac{1100}{x+10}$. Synthetic division is a powerful tool for simplifying complex polynomial expressions and is widely used in mathematics and engineering. We hope that this article has provided a clear understanding of how to perform synthetic division and has demonstrated its usefulness in solving polynomial division problems.

Final Answer

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Introduction

In our previous article, we used synthetic division to divide the polynomial $-x^3 + 69x - 310$ by $x + 10$ and obtained the result $-x^2 + 79 - \frac{1100}{x+10}$. Synthetic division is a powerful tool for simplifying complex polynomial expressions and is widely used in mathematics and engineering. In this article, we will answer some frequently asked questions about synthetic division.

Q&A

Q: What is synthetic division?

A: Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut method that eliminates the need for long division and is often used when dividing polynomials by factors of the form $(x - a)$ or $(x + a)$.

Q: How do I perform synthetic division?

A: To perform synthetic division, you need to follow these steps:

1. Write down the coefficients of the polynomial in a row, with the constant term on the right-hand side.
2. Write down the value of the linear factor in the middle of the row.
3. Multiply the value in the middle of the row by the first coefficient and write the result below the row.
4. Add the result from step 3 to the second coefficient and write the result below the row.
5. Repeat steps 3 and 4 until we reach the last coefficient.
6. The final result is the coefficients of the quotient polynomial.

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

A: Synthetic division is a shortcut method that eliminates the need for long division. It is often used when dividing polynomials by factors of the form $(x - a)$ or $(x + a)$. Long division, on the other hand, is a more general method that can be used to divide polynomials by any linear factor.

Q: Can I use synthetic division to divide polynomials by factors of the form $(x^2 + ax + b)$?

A: No, synthetic division can only be used to divide polynomials by linear factors of the form $(x - a)$ or $(x + a)$. If you need to divide a polynomial by a quadratic factor, you will need to use a different method, such as factoring or using the quadratic formula.

Q: How do I know if I have performed synthetic division correctly?

A: To check if you have performed synthetic division correctly, you can multiply the quotient polynomial by the linear factor and add the remainder. If the result is equal to the original polynomial, then you have performed synthetic division correctly.

Q: Can I use synthetic division to divide polynomials with complex coefficients?

A: Yes, synthetic division can be used to divide polynomials with complex coefficients. However, you will need to use complex numbers and follow the same steps as you would for real numbers.

Q: Is synthetic division only used in mathematics?

A: No, synthetic division is used in many fields, including engineering, physics, and computer science. It is a powerful tool for simplifying complex polynomial expressions and is widely used in many areas of mathematics and science.

Conclusion

In this article, we have answered some frequently asked questions about synthetic division. Synthetic division is a powerful tool for simplifying complex polynomial expressions and is widely used in mathematics and engineering. We hope that this article has provided a clear understanding of synthetic division and has demonstrated its usefulness in solving polynomial division problems.

Final Answer

The final answer is $\boxed{Yes, synthetic division is a powerful tool for simplifying complex polynomial expressions and is widely used in mathematics and engineering.}$