Use synthetic division to find the quotient and remainder when x4 − 9x2 + 3x + 6 is divided by x − 3.

Sure, let's use synthetic division to find the quotient and remainder when x^4 - 9x^2 + 3x + 6 is divided by x - 3.

Step 1: Write down the coefficients of the dividend (the polynomial we're dividing), which are 1 (for x^4), 0 (for x^3, which is missing from the polynomial), -9 (for x^2), 3 (for x), and 6 (the constant term).

Step 2: Write down the zero of the divisor (the polynomial we're dividing by). The zero is the value of x that makes the divisor equal to zero. In this case, the divisor is x - 3, so the zero is 3.

Step 3: Set up the synthetic division. Draw a line, write the zero on the left, and the coefficients of the dividend on the right:

3 | 1  0  -9  3  6

Step 4: Bring down the first coefficient (1), multiply it by the zero (3), and write the result under the second coefficient (0):

3 | 1  0  -9  3  6
   |    3
  -------------
      1  3

Step 5: Add the numbers in the second column (0 + 3), multiply the result by the zero (3), and write the result under the third coefficient (-9):

3 | 1  0  -9  3  6
   |    3  9
  -------------
      1  3  0

Step 6: Repeat this process for the remaining coefficients:

3 | 1  0  -9  3  6
   |    3  9  0  0
  -------------
      1  3  0  3  6

Step 7: The numbers on the bottom are the coefficients of the quotient and the remainder. The first number (1) is the coefficient of x^3, the second number (3) is the coefficient of x^2, the third number (0) is the coefficient of x, and the fourth number (3) is the constant term. The last number (6) is the remainder.

So, the quotient is x^3 + 3x^2 + 3 and the remainder is 6.