If $p(w) = 7w^3 - 40w^2 + 29w - 39$, Use Synthetic Division To Find $p(5)$.
Introduction
Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is often used to find the remainder of a polynomial when divided by a linear factor. In this article, we will use synthetic division to find the value of the polynomial $p(w) = 7w^3 - 40w^2 + 29w - 39$ when $w = 5$.
Understanding Synthetic Division
Synthetic division is a step-by-step process that involves dividing the coefficients of the polynomial by the linear factor. The process involves writing down the coefficients of the polynomial in a row, followed by the root of the linear factor. The coefficients are then divided by the root, and the result is written down below the row of coefficients.
Setting Up the Synthetic Division
To find $p(5)$, we need to set up the synthetic division with the coefficients of the polynomial and the root $w = 5$. The coefficients of the polynomial are $7, -40, 29, -39$, and the root is $5$.
Performing the Synthetic Division
To perform the synthetic division, we start by writing down the coefficients of the polynomial in a row, followed by the root.
| 7 | -40 | 29 | -39 | |
|---|---|---|---|---|
| 5 |
Next, we divide the first coefficient by the root and write down the result below the row of coefficients.
| 7 | -40 | 29 | -39 | |
|---|---|---|---|---|
| 5 | 1 |
Then, we multiply the result by the root and subtract it from the second coefficient.
| 7 | -40 | 29 | -39 | |
|---|---|---|---|---|
| 5 | 1 | -35 |
Next, we divide the result by the root and write down the result below the row of coefficients.
| 7 | -40 | 29 | -39 | |
|---|---|---|---|---|
| 5 | 1 | -35 | 14 |
Then, we multiply the result by the root and subtract it from the third coefficient.
| 7 | -40 | 29 | -39 | |
|---|---|---|---|---|
| 5 | 1 | -35 | 14 | -39 |
Next, we divide the result by the root and write down the result below the row of coefficients.
| 7 | -40 | 29 | -39 | |
|---|---|---|---|---|
| 5 | 1 | -35 | 14 | -39 |
Then, we multiply the result by the root and subtract it from the fourth coefficient.
| 7 | -40 | 29 | -39 | |
|---|---|---|---|---|
| 5 | 1 | -35 | 14 | -39 |
Finding the Value of the Polynomial
The final result of the synthetic division is the value of the polynomial when divided by the linear factor. In this case, the final result is $1 - 35 + 14 - 39 = -59$. Therefore, the value of the polynomial $p(w) = 7w^3 - 40w^2 + 29w - 39$ when $w = 5$ is $-59$.
Conclusion
In this article, we used synthetic division to find the value of the polynomial $p(w) = 7w^3 - 40w^2 + 29w - 39$ when $w = 5$. The process involved setting up the synthetic division with the coefficients of the polynomial and the root, performing the synthetic division, and finding the final result. The final result was $-59$, which is the value of the polynomial when $w = 5$.
Example Use Cases
Synthetic division is a useful tool for finding the remainder of a polynomial when divided by a linear factor. It is often used in algebra and calculus to find the value of a polynomial at a specific point. Some example use cases of synthetic division include:
- Finding the value of a polynomial at a specific point
- Finding the remainder of a polynomial when divided by a linear factor
- Solving systems of equations
- Finding the roots of a polynomial
Tips and Tricks
Here are some tips and tricks for using synthetic division:
- Make sure to set up the synthetic division correctly with the coefficients of the polynomial and the root.
- Perform the synthetic division step-by-step, making sure to write down the result of each step.
- Check the final result to make sure it is correct.
- Use synthetic division to find the value of a polynomial at a specific point, or to find the remainder of a polynomial when divided by a linear factor.
Common Mistakes
Here are some common mistakes to avoid when using synthetic division:
- Not setting up the synthetic division correctly with the coefficients of the polynomial and the root.
- Not performing the synthetic division step-by-step, or not writing down the result of each step.
- Not checking the final result to make sure it is correct.
- Not using synthetic division to find the value of a polynomial at a specific point, or to find the remainder of a polynomial when divided by a linear factor.
Conclusion
Synthetic division is a useful tool for finding the remainder of a polynomial when divided by a linear factor. It is often used in algebra and calculus to find the value of a polynomial at a specific point. By following the steps outlined in this article, you can use synthetic division to find the value of a polynomial at a specific point, or to find the remainder of a polynomial when divided by a linear factor.
Introduction
Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut to the long division method and is often used to find the remainder of a polynomial when divided by a linear factor. In this article, we will answer some common 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 often used to find the remainder of a polynomial when divided by a linear factor.
Q: How do I set up synthetic division?
A: To set up synthetic division, you need to write down the coefficients of the polynomial in a row, followed by the root of the linear factor.
Q: What is the root of the linear factor?
A: The root of the linear factor is the value that the polynomial is being divided by. For example, if you are dividing the polynomial by (x - 5), the root is 5.
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 in a row, followed by the root of the linear factor.
- Divide the first coefficient by the root and write down the result below the row of coefficients.
- Multiply the result by the root and subtract it from the second coefficient.
- Divide the result by the root and write down the result below the row of coefficients.
- Repeat steps 3 and 4 until you have finished dividing the polynomial.
Q: What is the final result of synthetic division?
A: The final result of synthetic division is the value of the polynomial when divided by the linear factor.
Q: Can I use synthetic division to find the roots of a polynomial?
A: Yes, you can use synthetic division to find the roots of a polynomial. If the remainder of the division is 0, then the root is a root of the polynomial.
Q: Can I use synthetic division to find the value of a polynomial at a specific point?
A: Yes, you can use synthetic division to find the value of a polynomial at a specific point. If you divide the polynomial by (x - a), the final result will be the value of the polynomial at x = a.
Q: What are some common mistakes to avoid when using synthetic division?
A: Some common mistakes to avoid when using synthetic division include:
- Not setting up the synthetic division correctly with the coefficients of the polynomial and the root.
- Not performing the synthetic division step-by-step, or not writing down the result of each step.
- Not checking the final result to make sure it is correct.
- Not using synthetic division to find the value of a polynomial at a specific point, or to find the remainder of a polynomial when divided by a linear factor.
Q: Can I use synthetic division to divide polynomials by quadratic factors?
A: No, you cannot use synthetic division to divide polynomials by quadratic factors. Synthetic division is only used to divide polynomials by linear factors.
Q: Can I use synthetic division to divide polynomials by rational factors?
A: No, you cannot use synthetic division to divide polynomials by rational factors. Synthetic division is only used to divide polynomials by linear factors.
Q: Can I use synthetic division to divide polynomials by complex factors?
A: Yes, you can use synthetic division to divide polynomials by complex factors. However, you need to use complex numbers to perform the division.
Q: Can I use synthetic division to divide polynomials by irrational factors?
A: No, you cannot use synthetic division to divide polynomials by irrational factors. Synthetic division is only used to divide polynomials by linear factors.
Conclusion
Synthetic division is a useful tool for finding the remainder of a polynomial when divided by a linear factor. It is often used in algebra and calculus to find the value of a polynomial at a specific point. By following the steps outlined in this article, you can use synthetic division to find the value of a polynomial at a specific point, or to find the remainder of a polynomial when divided by a linear factor.
Example Use Cases
Synthetic division is a useful tool for finding the remainder of a polynomial when divided by a linear factor. It is often used in algebra and calculus to find the value of a polynomial at a specific point. Some example use cases of synthetic division include:
- Finding the value of a polynomial at a specific point
- Finding the remainder of a polynomial when divided by a linear factor
- Solving systems of equations
- Finding the roots of a polynomial
Tips and Tricks
Here are some tips and tricks for using synthetic division:
- Make sure to set up the synthetic division correctly with the coefficients of the polynomial and the root.
- Perform the synthetic division step-by-step, making sure to write down the result of each step.
- Check the final result to make sure it is correct.
- Use synthetic division to find the value of a polynomial at a specific point, or to find the remainder of a polynomial when divided by a linear factor.
Common Mistakes
Here are some common mistakes to avoid when using synthetic division:
- Not setting up the synthetic division correctly with the coefficients of the polynomial and the root.
- Not performing the synthetic division step-by-step, or not writing down the result of each step.
- Not checking the final result to make sure it is correct.
- Not using synthetic division to find the value of a polynomial at a specific point, or to find the remainder of a polynomial when divided by a linear factor.