Use synthetic division to find P(–3) for P(x)=x^4–2x^3–4x+4 .

Answer 1

To find P(-3) using synthetic division, we will divide the polynomial P(x) by (x + 3).

First, we need to set up the synthetic division table:

-3 | 1 -2 0 -4 4

Since the leading coefficient is 1, we write it at the top of the division table:

-3 | 1 -2 0 -4 4
|__________________

Next, we bring down the first coefficient, which is 1:

-3 | 1 -2 0 -4 4
| 1
|__________________

Next, we multiply the -3 (the divisor) by 1 (the number we brought down) and write the result below the next coefficient:

-3 | 1 -2 0 -4 4
| 1
| -3
|__________________

Next, we add (-2) and (-3) to get -5, and write the result below the next coefficient:

-3 | 1 -2 0 -4 4
| 1 -5
| -3
|__________________

Next, we multiply the -3 (the divisor) by -5 (the number we just calculated) and write the result below the next coefficient:

-3 | 1 -2 0 -4 4
| 1 -5 15
| -3
|__________________

Next, we add 0 and 15 to get 15, and write the result below the next coefficient:

-3 | 1 -2 0 -4 4
| 1 -5 15
| -3 15
|__________________

Lastly, we multiply the -3 (the divisor) by 15 (the number we just calculated) and write the result below the last coefficient:

-3 | 1 -2 0 -4 4
| 1 -5 15 -33
| -3 15
|__________________

Finally, we add -4 and -33 to get -37, which is the remainder:

P(-3) = -37

Therefore, P(-3) = -37.