Use synthetic division and the Remainder Theorem to find P(-5) if P(x) = -x^3 - 4x^2 + x - 2.

Please help....?

p(x)=x^3+7x^2+4x a=-2

For the remainder theorem. Substitute (-5) for x and simplify.

For synthetic division:

-5| -1 -4 1 -2

To find P(-5) using synthetic division and the Remainder Theorem, follow these steps:

Step 1: Write down the coefficients of the polynomial P(x) in decreasing order of the powers of x. In this case, the polynomial is P(x) = -x^3 - 4x^2 + x - 2, so the coefficients are -1, -4, 1, and -2.

Step 2: Use synthetic division to divide the polynomial by (x - (-5)). In this case, the divisor is (x + 5). Set up the synthetic division as shown below:

-5 | -1 -4 1 -2
|_____

Step 3: Bring down the first coefficient (-1) to the bottom row.

-5 | -1 -4 1 -2
|_____
-5

Step 4: Multiply the divisor (-5) by the result (-5), and write the product (-25) below the next coefficient (-4). Then add the two values to get the new value (-29).

-5 | -1 -4 1 -2
|_____
-5
-1____
-4 -29

Step 5: Repeat the process by multiplying the divisor (-5) by the new result (-29), and write the product (145) below the next coefficient (1). Then add the two values to get the new value (146).

-5 | -1 -4 1 -2
|_____
-5
-1____
-4 -29
29_____
1 146

Step 6: Again, multiply the divisor (-5) by the new result (146), and write the product (-730) below the last coefficient (-2). Then add the two values to get the new value (-732).

-5 | -1 -4 1 -2
|_____
-5
-1____
-4 -29
29_____
1 146
-146____
-2 -732

Step 7: The last value obtained (-732) is the remainder of the division.

Therefore, P(-5) = -732.

Hence, the value of P(-5) using synthetic division is -732.

To find P(-5) using synthetic division and the Remainder Theorem, you can follow these steps:

Step 1: Set up the synthetic division.

In synthetic division, we will use the constant term and the coefficients of the polynomial to set up the division table.

The polynomial P(x) = -x^3 - 4x^2 + x - 2 can be rewritten with 0 as the coefficient of x when a term is missing:

P(x) = -x^3 - 4x^2 + x^1 + 0x^0 - 2

Now we can set up the synthetic division table:

| -1 -4 1 0 -2
-5 |____________________

Step 2: Perform the synthetic division.

To perform the synthetic division, we start by bringing down the first coefficient, which is -1, to the bottom row of the table:

| -1 -4 1 0 -2
-5 |-1

Next, we multiply the number at the bottom by the divisor, -5, and write the result below the next coefficient:

| -1 -4 1 0 -2
-5 |-1 5

Add the numbers in the second column (-4 + 5) and write the result below the line:

| -1 -4 1 0 -2
-5 |-1 5 1

Repeat the process until you reach the last column of the table:

| -1 -4 1 0 -2
-5 |-1 5 1 -5
______________
-1 1 2 -5

Step 3: Interpret the result.

The last number in the bottom row after performing the synthetic division, -5, is the remainder. According to the Remainder Theorem, this remainder is equal to P(-5).

Therefore, P(-5) = -5.

So, the value of P(-5) for the given polynomial is -5.