Use synthetic division to find P(−2) for P(x)=x4−5x3+3x+2 .
To find P(−2) for the polynomial P(x) = x^4 - 5x^3 + 3x + 2, we will use synthetic division.
1. Write down the coefficients of the polynomial in descending order:
1, -5, 0, 3, 2
2. Set up the synthetic division with -2 as the divisor:
-2 | 1 -5 0 3 2
3. Bring down the first coefficient:
-2 | 1 -5 0 3 2
1
4. Multiply -2 by 1 and add to the second coefficient:
-2 | 1 -5 0 3 2
1 2
5. Multiply -2 by 2 and add to the third coefficient:
-2 | 1 -5 0 3 2
1 2 4
6. Multiply -2 by 4 and add to the fourth coefficient:
-2 | 1 -5 0 3 2
1 2 4 -2
7. Multiply -2 by -2 and add to the fifth coefficient:
-2 | 1 -5 0 3 2
1 2 4 -2 6
8. The last number in the synthetic division is the remainder, which is the result of P(-2):
P(-2) = 6
Therefore, P(-2) = 6 for the polynomial P(x) = x^4 - 5x^3 + 3x + 2.