Answer :
$\bullet$ Set up the synthetic division with the coefficients of the dividend and the root of the divisor.
$\bullet$ Perform the synthetic division process.
$\bullet$ Identify the coefficients of the quotient and the remainder.
$\bullet$ Write the quotient and remainder: The result of the division is $\boxed{4x^3 - 3x^2 + x - 1}$.
### Explanation
1. Understanding the problem
We are asked to divide the polynomial $4 x^4-19 x^3+13 x^2-5 x+4$ by $x-4$. This can be done using polynomial long division or synthetic division. Here, we will use synthetic division.
2. Setting up synthetic division
To set up synthetic division, we write down the coefficients of the dividend polynomial $4 x^4-19 x^3+13 x^2-5 x+4$, which are 4, -19, 13, -5, and 4. The divisor is $x-4$, so we use 4 as the divisor in synthetic division.
3. Performing synthetic division
Now, we perform the synthetic division:
1. Bring down the first coefficient (4).
2. Multiply the divisor (4) by the number we brought down (4) to get 16.
3. Add the result (16) to the next coefficient (-19) to get -3.
4. Multiply the divisor (4) by -3 to get -12.
5. Add the result (-12) to the next coefficient (13) to get 1.
6. Multiply the divisor (4) by 1 to get 4.
7. Add the result (4) to the next coefficient (-5) to get -1.
8. Multiply the divisor (4) by -1 to get -4.
9. Add the result (-4) to the last coefficient (4) to get 0.
The numbers we obtained in the process are 4, -3, 1, -1, and 0.
4. Determining the quotient and remainder
The numbers 4, -3, 1, and -1 are the coefficients of the quotient polynomial, and the last number 0 is the remainder. Therefore, the quotient is $4x^3 - 3x^2 + x - 1$ and the remainder is 0.
5. Final result
Thus, the result of the division is $4x^3 - 3x^2 + x - 1$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as control systems, signal processing, and structural analysis. For example, engineers use polynomial division to analyze the stability of control systems by finding the roots of the characteristic equation. In signal processing, it helps in designing digital filters. Understanding polynomial division is also crucial in computer graphics for tasks like curve fitting and surface modeling.
$\bullet$ Perform the synthetic division process.
$\bullet$ Identify the coefficients of the quotient and the remainder.
$\bullet$ Write the quotient and remainder: The result of the division is $\boxed{4x^3 - 3x^2 + x - 1}$.
### Explanation
1. Understanding the problem
We are asked to divide the polynomial $4 x^4-19 x^3+13 x^2-5 x+4$ by $x-4$. This can be done using polynomial long division or synthetic division. Here, we will use synthetic division.
2. Setting up synthetic division
To set up synthetic division, we write down the coefficients of the dividend polynomial $4 x^4-19 x^3+13 x^2-5 x+4$, which are 4, -19, 13, -5, and 4. The divisor is $x-4$, so we use 4 as the divisor in synthetic division.
3. Performing synthetic division
Now, we perform the synthetic division:
1. Bring down the first coefficient (4).
2. Multiply the divisor (4) by the number we brought down (4) to get 16.
3. Add the result (16) to the next coefficient (-19) to get -3.
4. Multiply the divisor (4) by -3 to get -12.
5. Add the result (-12) to the next coefficient (13) to get 1.
6. Multiply the divisor (4) by 1 to get 4.
7. Add the result (4) to the next coefficient (-5) to get -1.
8. Multiply the divisor (4) by -1 to get -4.
9. Add the result (-4) to the last coefficient (4) to get 0.
The numbers we obtained in the process are 4, -3, 1, -1, and 0.
4. Determining the quotient and remainder
The numbers 4, -3, 1, and -1 are the coefficients of the quotient polynomial, and the last number 0 is the remainder. Therefore, the quotient is $4x^3 - 3x^2 + x - 1$ and the remainder is 0.
5. Final result
Thus, the result of the division is $4x^3 - 3x^2 + x - 1$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as control systems, signal processing, and structural analysis. For example, engineers use polynomial division to analyze the stability of control systems by finding the roots of the characteristic equation. In signal processing, it helps in designing digital filters. Understanding polynomial division is also crucial in computer graphics for tasks like curve fitting and surface modeling.
