Final answer:
To find the remainder using synthetic division, divide the polynomial 3x⁴+10x³-6x²+5x-7 by x+4 using synthetic division.
Step-by-step explanation:
To find the remainder using synthetic division, we need to first set up the division by putting the polynomial in the proper form. The polynomial is 3x⁴+10x³-6x²+5x-7 and we are dividing by x+4. Write the coefficients of the polynomial in descending order of exponents:
3x⁴+10x³-6x²+5x-7 = 3, 10, -6, 5, -7
Using synthetic division, bring down the first coefficient:
3
Next, multiply the divisor (x+4) by the number at the top and add the result to the next coefficient:
3, 10+3=13
Repeat this process until all coefficients have been divided:
3, 13, 7, 27, 101
The last number represents the remainder. In this case, the remainder is 101.