Use Synthetic Division To Find The Result When $2x^3 - 7x^2 + 12x - 7$ Is Divided By $x - 1$.

Mar 12, 2025 by ADMIN 98 views

**Synthetic Division: A Powerful Tool for Polynomial Division**

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). In this article, we will use synthetic division to find the result when the polynomial $2x^3 - 7x^2 + 12x - 7$ is divided by $x - 1$.

What is Synthetic Division?

Synthetic division is a method of dividing polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). The method involves setting up a table with the coefficients of the polynomial and the root of the divisor. The table is then used to calculate the coefficients of the quotient and the remainder.

How to Perform Synthetic Division

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

Write down the coefficients of the polynomial: Write down the coefficients of the polynomial in a row, starting with the coefficient of the highest degree term.
Write down the root of the divisor: Write down the root of the divisor, which is the value of x that makes the divisor equal to zero.
Bring down the first coefficient: Bring down the first coefficient of the polynomial.
Multiply the root by the first coefficient: Multiply the root by the first coefficient and write the result below the second coefficient.
Add the second coefficient and the result: Add the second coefficient and the result from step 4.
Repeat steps 4 and 5: Repeat steps 4 and 5 for each coefficient, multiplying the root by the previous result and adding the next coefficient.
The final result is the quotient and remainder: The final result is the quotient and remainder.

Example: Dividing $2x^3 - 7x^2 + 12x - 7$ by $x - 1$

Now, let's use synthetic division to find the result when the polynomial $2x^3 - 7x^2 + 12x - 7$ is divided by $x - 1$.

Step 1: Write down the coefficients of the polynomial

The coefficients of the polynomial are 2, -7, 12, and -7.

Step 2: Write down the root of the divisor

The root of the divisor is 1.

Step 3: Bring down the first coefficient

Bring down the first coefficient, which is 2.

Step 4: Multiply the root by the first coefficient

Multiply the root by the first coefficient and write the result below the second coefficient.

	2	-7	12	-7
1	2

Step 5: Add the second coefficient and the result

Add the second coefficient and the result from step 4.

	2	-7	12	-7
1	2	-5

Step 6: Repeat steps 4 and 5

Repeat steps 4 and 5 for each coefficient.

	2	-7	12	-7
1	2	-5	17

Step 7: The final result is the quotient and remainder

The final result is the quotient and remainder.

The quotient is $2x^2 - 5x + 17$ and the remainder is 0.

Conclusion

Synthetic division is a powerful tool for polynomial division. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). In this article, we used synthetic division to find the result when the polynomial $2x^3 - 7x^2 + 12x - 7$ is divided by $x - 1$. The quotient is $2x^2 - 5x + 17$ and the remainder is 0.

Applications of Synthetic Division

Synthetic division has many applications in mathematics and science. It is used to find the roots of polynomials, which is essential in many areas of mathematics and science. It is also used to find the quotient and remainder of polynomial division, which is essential in many areas of mathematics and science.

Limitations of Synthetic Division

Synthetic division has some limitations. It is only used to divide polynomials by linear factors of the form (x - a). It is not used to divide polynomials by quadratic factors or higher degree factors.

Future Research

There is ongoing research in the area of synthetic division. Researchers are working to develop new methods and algorithms for synthetic division. They are also working to apply synthetic division to new areas of mathematics and science.

Conclusion

In conclusion, synthetic division is a powerful tool for polynomial division. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). It has many applications in mathematics and science and is used to find the roots of polynomials, the quotient and remainder of polynomial division, and many other things. However, it has some limitations and is only used to divide polynomials by linear factors of the form (x - a).

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a). In this article, we will answer some frequently asked questions about synthetic division.

Q: What is synthetic division?

A: Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is particularly useful when the divisor is of the form (x - a).

Q: How do I perform synthetic division?

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

Write down the coefficients of the polynomial.
Write down the root of the divisor.
Bring down the first coefficient.
Multiply the root by the first coefficient and write the result below the second coefficient.
Add the second coefficient and the result.
Repeat steps 4 and 5 for each coefficient.
The final result is the quotient and remainder.

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

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

Q: How do I find the root of the divisor?

A: To find the root of the divisor, you need to set the divisor equal to zero and solve for x.

Q: What are the applications of synthetic division?

A: Synthetic division has many applications in mathematics and science. It is used to find the roots of polynomials, which is essential in many areas of mathematics and science. It is also used to find the quotient and remainder of polynomial division, which is essential in many areas of mathematics and science.

Q: What are the limitations of synthetic division?

A: Synthetic division has some limitations. It is only used to divide polynomials by linear factors of the form (x - a). It is not used to divide polynomials by quadratic factors or higher degree factors.

Q: Can I use synthetic division to divide polynomials by quadratic factors?

A: No, synthetic division is only used to divide polynomials by linear factors of the form (x - a). It is not used to divide polynomials by quadratic factors or higher degree factors.

Q: How do I use synthetic division to find the roots of a polynomial?

A: To use synthetic division to find the roots of a polynomial, you need to divide the polynomial by a linear factor of the form (x - a) and then set the divisor equal to zero and solve for x.

Q: What are some common mistakes to avoid when using synthetic division?

A: Some common mistakes to avoid when using synthetic division include:

Not bringing down the first coefficient.
Not multiplying the root by the first coefficient.
Not adding the second coefficient and the result.
Not repeating steps 4 and 5 for each coefficient.
Not checking the final result for errors.

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 need to be careful when working with complex numbers and make sure to follow the correct procedures.

Q: How do I use synthetic division to divide polynomials with rational coefficients?

A: To use synthetic division to divide polynomials with rational coefficients, you need to follow the same steps as for polynomials with integer coefficients. However, you need to be careful when working with rational numbers and make sure to follow the correct procedures.