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.

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, since there is no x^3 term), -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 in this case):

3 | 1
  |---
  | 1

Step 5: Multiply the zero by the number you just wrote down, and write the result under the next coefficient. Then add the numbers in that column and write the result below the line:

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

Repeat this process for the rest of the coefficients:

3 | 1  0  -9  3  6
  |   3  9  0  9
  |---
  | 1  3  0  3  15

Step 6: The numbers on the bottom line are the coefficients of the quotient and the remainder. The first number is the coefficient of x^3, the second is the coefficient of x^2, the third is the coefficient of x, and the fourth is the constant term. The last number is the remainder. So, the quotient is x^3 + 3x^2 + 3 and the remainder is 15.