Use Synthetic Division To Find The Quotient And Remainder When $5x^4 - 17x^3 - 18x^2 + 19x + 20$ Is Divided By $x - 4$ By Completing The Parts Below:(a) Complete This Synthetic Division Table.$\[ \begin{array}{rrrrr} \hline 4) & 5
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 dividing polynomials by factors of the form (x - a). In this article, we will use synthetic division to find the quotient and remainder when the polynomial is divided by .
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 dividing polynomials by factors of the form (x - a). The method involves setting up a table with the coefficients of the polynomial and the root of the linear factor. The table is then filled in using a series of steps, and the quotient and remainder are obtained.
How to Perform Synthetic Division
To perform synthetic division, we need to set up a table with the coefficients of the polynomial and the root of the linear factor. The table should have the following format:
| 1 | 5 | -17 | -18 | 19 | 20 | |
|---|---|---|---|---|---|---|
| 4 |
The first column of the table contains the coefficients of the polynomial, and the second column contains the root of the linear factor. The remaining columns are filled in using a series of steps.
Step 1: Write Down the Coefficients
The first step in performing synthetic division is to write down the coefficients of the polynomial. In this case, the coefficients are 5, -17, -18, 19, and 20.
Step 2: Bring Down the First Coefficient
The next step is to bring down the first coefficient, which is 5.
Step 3: Multiply the Root by the First Coefficient
The next step is to multiply the root, which is 4, by the first coefficient, which is 5. This gives us 20.
Step 4: Add the Product to the Second Coefficient
The next step is to add the product, which is 20, to the second coefficient, which is -17. This gives us 3.
Step 5: Multiply the Root by the Result
The next step is to multiply the root, which is 4, by the result, which is 3. This gives us 12.
Step 6: Add the Product to the Third Coefficient
The next step is to add the product, which is 12, to the third coefficient, which is -18. This gives us -6.
Step 7: Multiply the Root by the Result
The next step is to multiply the root, which is 4, by the result, which is -6. This gives us -24.
Step 8: Add the Product to the Fourth Coefficient
The next step is to add the product, which is -24, to the fourth coefficient, which is 19. This gives us -5.
Step 9: Multiply the Root by the Result
The next step is to multiply the root, which is 4, by the result, which is -5. This gives us -20.
Step 10: Add the Product to the Fifth Coefficient
The next step is to add the product, which is -20, to the fifth coefficient, which is 20. This gives us 0.
The Quotient and Remainder
The final step in performing synthetic division is to write down the quotient and remainder. The quotient is the polynomial obtained by dividing the original polynomial by the linear factor, and the remainder is the constant term obtained in the final step.
In this case, the quotient is 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 dividing polynomials by factors of the form (x - a). By following the steps outlined in this article, we can use synthetic division to find the quotient and remainder when a polynomial is divided by a linear factor.
Example
Let's use synthetic division to find the quotient and remainder when the polynomial is divided by .
| 1 | 1 | 2 | 3 | 4 | |
|---|---|---|---|---|---|
| -1 |
The first step is to write down the coefficients of the polynomial, which are 1, 2, 3, and 4.
The next step is to bring down the first coefficient, which is 1.
The next step is to multiply the root, which is -1, by the first coefficient, which is 1. This gives us -1.
The next step is to add the product, which is -1, to the second coefficient, which is 2. This gives us 1.
The next step is to multiply the root, which is -1, by the result, which is 1. This gives us -1.
The next step is to add the product, which is -1, to the third coefficient, which is 3. This gives us 2.
The next step is to multiply the root, which is -1, by the result, which is 2. This gives us -2.
The next step is to add the product, which is -2, to the fourth coefficient, which is 4. This gives us 2.
The final step is to write down the quotient and remainder. The quotient is and the remainder is 2.
Applications of Synthetic Division
Synthetic division has many applications in mathematics and science. Some of the most common applications include:
- Polynomial division: Synthetic division is used to divide polynomials by linear factors.
- Root finding: Synthetic division is used to find the roots of polynomials.
- Graphing: Synthetic division is used to graph polynomials.
- Numerical analysis: Synthetic division is used in numerical analysis to solve systems of equations.
Conclusion
Frequently Asked Questions
Q: What is synthetic division?
A: 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 dividing polynomials by factors of the form (x - a).
Q: How do I perform synthetic division?
A: To perform synthetic division, you need to set up a table with the coefficients of the polynomial and the root of the linear factor. The table should have the following format:
| 1 | 5 | -17 | -18 | 19 | 20 | |
|---|---|---|---|---|---|---|
| 4 |
The first column of the table contains the coefficients of the polynomial, and the second column contains the root of the linear factor. The remaining columns are filled in using a series of steps.
Q: What are the steps involved in synthetic division?
A: The steps involved in synthetic division are:
- Write down the coefficients of the polynomial.
- Bring down the first coefficient.
- Multiply the root by the first coefficient.
- Add the product to the second coefficient.
- Multiply the root by the result.
- Add the product to the third coefficient.
- Multiply the root by the result.
- Add the product to the fourth coefficient.
- Multiply the root by the result.
- Add the product to the fifth coefficient.
Q: What is the quotient and remainder in synthetic division?
A: The quotient is the polynomial obtained by dividing the original polynomial by the linear factor, and the remainder is the constant term obtained in the final step.
Q: How do I find the roots of a polynomial using synthetic division?
A: To find the roots of a polynomial using synthetic division, you need to set up a table with the coefficients of the polynomial and the root of the linear factor. The table should have the following format:
| 1 | 5 | -17 | -18 | 19 | 20 | |
|---|---|---|---|---|---|---|
| 4 |
The first column of the table contains the coefficients of the polynomial, and the second column contains the root of the linear factor. The remaining columns are filled in using a series of steps.
Q: Can I use synthetic division to divide polynomials by factors of the form (x + a)?
A: Yes, you can use synthetic division to divide polynomials by factors of the form (x + a). The only difference is that you need to use the negative of the root in the table.
Q: What are the applications of synthetic division?
A: Synthetic division has many applications in mathematics and science, including:
- Polynomial division: Synthetic division is used to divide polynomials by linear factors.
- Root finding: Synthetic division is used to find the roots of polynomials.
- Graphing: Synthetic division is used to graph polynomials.
- Numerical analysis: Synthetic division is used in numerical analysis to solve systems of equations.
Q: Is synthetic division a shortcut to the long division method?
A: Yes, synthetic division is a shortcut to the long division method. It is particularly useful when dividing polynomials by factors of the form (x - a).
Q: Can I use synthetic division to divide polynomials by factors of the form (x^2 + ax + b)?
A: No, you cannot use synthetic division to divide polynomials by factors of the form (x^2 + ax + b). Synthetic division is only used to divide polynomials by linear factors.
Conclusion
Synthetic division is a powerful tool for polynomial division. It is a shortcut to the long division method and is particularly useful when dividing polynomials by factors of the form (x - a). By following the steps outlined in this article, you can use synthetic division to find the quotient and remainder when a polynomial is divided by a linear factor. Synthetic division has many applications in mathematics and science, including polynomial division, root finding, graphing, and numerical analysis.