Use synthetic division to find P(-2) for p(x) = x^4-5x^3+3x+2 show all your steps to help get this answer in detail
To find P(-2) for the polynomial p(x) = x^4 - 5x^3 + 3x + 2 using synthetic division, follow these steps:
1. Write down the coefficients of the polynomial in descending order, filling in any missing terms with a coefficient of 0 if necessary:
p(x) = 1x^4 - 5x^3 + 0x^2 + 3x + 2
2. Set up the synthetic division with the value -2 outside the box and the coefficients inside the box:
-2 | 1 -5 0 3 2
3. Bring down the first coefficient (1) below the division bar:
-2 | 1 -5 0 3 2
1
4. Multiply the number outside the box (-2) by the number you just brought down (1) and write the result below the next coefficient:
-2 | 1 -5 0 3 2
1
-2
5. Add the two numbers together in the next column:
-2 | 1 -5 0 3 2
1
-2
-7
6. Repeat steps 4 and 5 for the rest of the coefficients:
-2 | 1 -5 0 3 2
1
-2
-7
14
-1
7. The final value below the division bar is the remainder, which is the value of P(-2):
P(-2) = -1
Therefore, P(-2) for the polynomial p(x) = x^4 - 5x^3 + 3x + 2 is equal to -1.