Answer :
Using synthetic division, the polynomial 6x⁵+16x⁴+13x³+23x²+24x-4 divided by x+2 results in a quotient of 6x⁴ + 4x³ + 5x² + 13x - 2 with a remainder of -40.
We need to use synthetic division to divide the polynomial 6x⁵+16x⁴+13x³+23x²+24x-4 by x+2.
Let's set up the synthetic division:
- First, we write down the coefficients of the dividend: 6, 16, 13, 23, 24, -4.
- We use -2 as our synthetic number because we are dividing by x + 2.
- Bring down the first coefficient, 6, to the bottom row.
- Multiply -2 by 6 and write the result, which is -12, under the second coefficient 16. Then add these two numbers to get 4. This is the new second coefficient for the bottom row.
- Continue this process: multiply -2 by the new second coefficient and add to the next coefficient, until you reach the end.
The results give us the coefficients of the quotient polynomial. The final row, after carrying out these calculations, will have the coefficients of the quotient. If there's a non-zero remainder, it will be the last number in that row.
So, we do the calculation and give final answer:
- -2 | 6 16 13 23 24 -4
- | -12 -8 -10 -26 -36
- ---------------------------
- 6 4 5 13 -2 -40
The quotient is 6x⁴ + 4x³ + 5x² + 13x - 2 and the remainder is -40.
So, when divided, the polynomial 6x⁵+16x⁴+13x³+23x²+24x-4 over x+2 equals 6x⁴ + 4x³ + 5x² + 13x - 2 with a remainder of -40.
