Use synthetic division to find P(3) for P(x)=x^4-6x^3-4x^2-6x-2

Can someone please explain

hard to explain synthetic division here. If you know how to do it, then you just need to know that

p(3) is the remainder after doing the division.

That's because p(x) = (x-3)*q(x) + p(3)

If x-3 divides p(x), then there is no remainder: p(3) = 0.

To use synthetic division to find P(3) for the given polynomial P(x), follow these steps:

Step 1: Write down the coefficients of the polynomial in descending order.
P(x) = x^4 - 6x^3 - 4x^2 - 6x - 2

Coefficients: 1, -6, -4, -6, -2

Step 2: Set up the synthetic division table, using the divisor as the number inside the parentheses and the coefficients of the polynomial as the first row.

3 | 1 -6 -4 -6 -2

Step 3: Bring down the first coefficient, which is 1.

3 | 1 -6 -4 -6 -2
____________________


Step 4: Multiply the divisor (3) by the carried down number (1), and write the result below the next coefficient (-6). Add these two numbers together and write the sum in the next column.

3 | 1 -6 -4 -6 -2
____________________
3

3 | 1 -6 -4 -6 -2
____________________
3 -9

Step 5: Repeat the process for the next column. Multiply the divisor (3) by the new carried down number (-9), and write the result below the next coefficient (-4). Add these two numbers together and write the sum in the next column.

3 | 1 -6 -4 -6 -2
____________________
3 -9
_____________
3

3 | 1 -6 -4 -6 -2
____________________
3 -9 -1

Step 6: Repeat the process for the next column, and continue until you reach the last column.

3 | 1 -6 -4 -6 -2
____________________
3 -9 -1
_____________
3 -6
_____________
2

3 | 1 -6 -4 -6 -2
____________________
3 -9 -1
_____________
3 -6
_____________
2 6

Step 7: The last number in the final row (-2) is the remainder.

Step 8: The values in the next row from left to right represent the coefficients of the quotient polynomial. In this case, it is 3x^3 - 6x^2 + 2x + 6.

Step 9: Therefore, P(3) = -2.

To summarize, the synthetic division method helps us find the remainder when dividing a polynomial by a linear factor (divisor). In this case, the remainder is -2, which is equivalent to P(3) for the given polynomial P(x).