Use synthetic division to determine whether \( x-c \) is a factor of the given polynomial. \( 4 x^{4}-20 x^{3}-6 x+30 ; x-5 \) Is \( x-5 \) a factor of the given polynomial? No Yes

To determine whether x – 5 is a factor of the polynomial 4x⁴ – 20x³ – 6x + 30, we can use synthetic division with c = 5. Step 1. Write the coefficients of the polynomial. Note that the x² term is missing, so its coefficient is 0:   4  (x⁴), -20  (x³), 0  (x²), -6  (x), 30  (constant). Step 2. Set up the synthetic division with 5:      ________________________ 5 | 4   -20   0   -6  30 Step 3. Perform the synthetic division: 1. Bring down the first coefficient: 4. 2. Multiply 4 by 5 to get 20. Write 20 under the -20. 3. Add -20 and 20 to get 0. 4. Multiply 0 by 5 to get 0. Write 0 under the 0. 5. Add 0 and 0 to get 0. 6. Multiply 0 by 5 to get 0. Write 0 under -6. 7. Add -6 and 0 to get -6. 8. Multiply -6 by 5 to get -30. Write -30 under 30. 9. Add 30 and -30 to get 0. The complete synthetic division table should look like:          4   -20   0   -6  30 5 |             20    0   0   -30          ___________________________          4    0    0   -6   0 The remainder is 0. Step 4. Conclude that since the remainder is 0, x – 5 is a factor of the polynomial. Answer: Yes, x – 5 is a factor of 4x⁴ – 20x³ – 6x + 30.