Question

use synthetic division to divide (x^3 + 2x^2 - 12) by (x - 2)

the quotient is:
the remainder is:

Explanation:

🆕 New Concept Discovered: Synthetic Division
A shortcut method to divide polynomials by linear factors.

Step 1: Set up the synthetic division table

To divide \( x^3 + 2x^2 - 12 \) by \( x - 2 \), we first identify the divisor's root and the coefficients of the dividend.

1. Find the root of the divisor: Set \( x - 2 = 0 \), which gives \( x = 2 \). This value goes in the box on the left.
2. List the coefficients of the dividend: Write down the coefficients of \( x^3 + 2x^2 + 0x - 12 \). Note that we must include a \( 0 \) placeholder for the missing \( x \)-term:

\[
1 \quad 2 \quad 0 \quad -12
\]

Our setup looks like this:

  2 |   1    2    0   -12
    |
    ---------------------

---

Step 2: Perform the synthetic division steps

We follow a pattern of bringing down, multiplying by the root (2), and adding columns:

1. Bring down the first coefficient, \( 1 \):

     2 |   1    2    0   -12
       |
       ---------------------
           1

1. Multiply \( 1 \times 2 = 2 \), write it under the second coefficient, and add:

\[
2 + 2 = 4
\]

     2 |   1    2    0   -12
       |        2
       ---------------------
           1    4

1. Multiply \( 4 \times 2 = 8 \), write it under the third coefficient, and add:

\[
0 + 8 = 8
\]

     2 |   1    2    0   -12
       |        2    8
       ---------------------
           1    4    8

1. Multiply \( 8 \times 2 = 16 \), write it under the fourth coefficient, and add:

\[
-12 + 16 = 4
\]

     2 |   1    2    0   -12
       |        2    8    16
       ---------------------
           1    4    8  |  4

---

Step 3: Interpret the results

The numbers at the bottom row represent the coefficients of the quotient and the remainder:

• The bottom row is: \( 1 \quad 4 \quad 8 \quad | \quad 4 \)
• The last number, \( 4 \), is the remainder.
• The remaining numbers, \( 1, 4, 8 \), represent the coefficients of the quotient polynomial, which starts one degree lower than the original dividend (degree 2 instead of 3):

\[
1x^2 + 4x + 8
\]

Answer:

• Synthetic Division Table Values:
• Row 2 values (from left to right): 2, 8, 16
• Row 3 values (from left to right): 1, 4, 8, 4
• The quotient is: \( x^2 + 4x + 8 \)
• The remainder is: \( 4 \)