Q. Use synthetic division to divide \(3x^3 – 11x^2 + 16x – 30\) by \(x – 3\).

Synthetic division of the polynomial by x−3 — step-by-step

1. Write the polynomial and identify the divisor root.The dividend is \(3x^{3}-11x^{2}+16x-30\). The divisor is \(x-3\), whose root is \(3\).
2. List the coefficients and set up the synthetic division table.The coefficients of the dividend (from highest degree to constant) are 3, −11, 16, −30. Place the root 3 to the left and the coefficients in a row:

   3 |  3   -11    16   -30
     |       9    -6    30
     ---------------------
        3    -2    10     0
   

   The numbers that will be written on the lower row are produced by the synthetic division steps shown next.

3. Step-by-step operations (each operation explained separately).
   1. Bring down the first coefficient.Bring down the leading coefficient 3 to the bottom row. This becomes the first coefficient of the quotient.

      Bottom row first entry: \(3\).

   2. Multiply and add to get the next bottom entry.Multiply the bottom entry 3 by the root 3: \(3 \times 3 = 9\).

      Add this product to the next coefficient in the top row: \(-11 + 9 = -2\).

      Bottom row second entry: \(-2\).

   3. Repeat multiply and add.Multiply the new bottom entry \(-2\) by the root 3: \(-2 \times 3 = -6\).

      Add to the next coefficient: \(16 + (-6) = 10\).

      Bottom row third entry: \(10\).

   4. Final multiply and add to obtain the remainder.Multiply the bottom entry 10 by the root 3: \(10 \times 3 = 30\).

      Add to the final coefficient: \(-30 + 30 = 0\).

      The final bottom entry is the remainder: \(0\).

4. Form the quotient and state the remainder.The bottom row entries (except the final remainder) are the coefficients of the quotient polynomial, in descending powers of x. They are \(3\), \(-2\), and \(10\), which correspond to the quotient

   \(3x^{2} – 2x + 10\).

   The remainder is \(0\).

   Therefore, dividing \(3x^{3}-11x^{2}+16x-30\) by \(x-3\) yields the quotient \(3x^{2}-2x+10\) with remainder \(0\).

5. Optional check (value at the root).Evaluate the polynomial at \(x=3\) to confirm the remainder is zero:

   \(f(3)=3(3)^{3}-11(3)^{2}+16(3)-30=81-99+48-30=0\).

   This confirms the remainder is \(0\) and that \(x-3\) is a factor.

Final result: Quotient is \(3x^{2}-2x+10\) and remainder is \(0\).