Use synthetic division to divide the polynomials. \[ \frac{4 x^{5}-29 x^{4}+124 x^{3}-348 x^{2}-237 x+10}{x-5}= \]

To perform synthetic division for the polynomial \( \frac{4x^{5} - 29x^{4} + 124x^{3} - 348x^{2} - 237x + 10}{x - 5} \), we will use \( 5 \) (the zero of \( x - 5 \)). 1. Write down the coefficients of the polynomial: \( 4, -29, 124, -348, -237, 10 \). 2. Set up synthetic division. Place \( 5 \) on the left and the coefficients on the right. ``` _______________________ 5 | 4 -29 124 -348 -237 10 ``` 3. Bring down the first coefficient \( 4 \). ``` _______________________ 5 | 4 -29 124 -348 -237 10 | |________ 4 ``` 4. Multiply \( 5 \) by \( 4 \) and write the result under the next coefficient: \( 20 \). ``` _______________________ 5 | 4 -29 124 -348 -237 10 | 20 |________ 4 -9 ``` 5. Add \( -29 \) and \( 20 \) to get \( -9 \). Repeat this process: - Multiply \( 5 \) by \( -9 \): \( -45 \) - Add \( 124 - 45 = 79 \) ``` _______________________ 5 | 4 -29 124 -348 -237 10 | 20 -45 |________ 4 -9 79 ``` - Multiply \( 5 \) by \( 79 \): \( 395 \) - Add \( -348 + 395 = 47 \) ``` _______________________ 5 | 4 -29 124 -348 -237 10 | 20 -45 395 |________ 4 -9 79 47 ``` - Multiply \( 5 \) by \( 47 \): \( 235 \) - Add \( -237 + 235 = -2 \) ``` _______________________ 5 | 4 -29 124 -348 -237 10 | 20 -45 395 235 |________ 4 -9 79 47 -2 ``` - Multiply \( 5 \) by \( -2 \): \( -10 \) - Add \( 10 - 10 = 0 \) ``` _______________________ 5 | 4 -29 124 -348 -237 10 | 20 -45 395 -10 |________ 4 -9 79 47 -2 0 ``` Now, we conclude that the quotient is: \[ 4x^{4} - 9x^{3} + 79x^{2} + 47x - 2 \] And the remainder is \( 0 \). Thus, the final result of the synthetic division is: \[ \frac{4x^5 - 29x^4 + 124x^3 - 348x^2 - 237x + 10}{x - 5} = 4x^{4} - 9x^{3} + 79x^{2} + 47x - 2 \]