How can we use synthetic division to find p(2) for the polynomial p(x) = x^4 + 9x^3 + 9x^2?

Synthetic division is a simplified form of polynomial long division, and it can be particularly useful for evaluating polynomials at specific points. To find p(2) for the polynomial p(x) = x⁴ + 9x³ + 9x², we will perform synthetic division using 2 as the divisor.

First, we need to set up our synthetic division:

1. Write down the coefficients of the polynomial. For p(x) = x⁴ + 9x³ + 9x² + 0x + 0, the coefficients are: [1, 9, 9, 0, 0].
2. Place the divisor (2) to the left.

   2 | 1   9   9   0   0  
     |

Now, we will proceed with synthetic division:

1. Bring down the leading coefficient (1):

   2 | 1   9   9   0   0  
     |        
     | 1

1. Multiply 2 by 1 and write the result under the next coefficient:

   2 | 1   9   9   0   0  
     |      2  
     | 1   11

1. Add the column to get 11.

   2 | 1   9   9   0   0  
     |      2  
     | 1   11  

1. Repeat this process:

   2 | 1   9   9   0   0  
     |      2   22  
     | 1  11  31  

We get:

      2 | 1  9  9  0  0  
      |   2  22  62  
      | 1  11  31  62

1. Finally, add the last column:

   2 | 1   9   9   0   0  
     |      2   22   62  
     |  1  11  31  62  

This means that the remainder is 62.

Therefore, p(2) = 62. This shows that using synthetic division not only helps us evaluate the polynomial but also streamlines the process compared to traditional long division methods.