Question

use synthetic division to divide the polynomial by the two known zeros, starting with 7i\
step 3\
use synthetic division again to divide the second known zero (the conjugate) into the quotient from the last step. use the first three numbers in the third row as the coefficients of the current dividend. (notice that after dividing just one complex zero, some coefficients of the resulting quotient were complex.)\
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Explanation:

First, we identify the initial polynomial and the first synthetic division result. From the completed step, the quotient polynomial after dividing by $7i$ is $x^2 + (1+7i)x + 7i$ (from the third row coefficients $1, 1+7i, 7i, 0$). The conjugate of $7i$ is $-7i$, which we use for the second synthetic division.

Step1: Set up synthetic division

Divisor: $-7i$, coefficients: $1, 1+7i, 7i$

-7i |  1    1+7i    7i

Step2: Bring down leading coefficient

Bring down 1.

-7i |  1    1+7i    7i
          ↓
        1

Step3: Multiply and add

Multiply $1 \times (-7i) = -7i$, add to $1+7i$:
$1+7i + (-7i) = 1$

-7i |  1    1+7i    7i
          ↓   -7i
        1      1

Step4: Multiply and add

Multiply $1 \times (-7i) = -7i$, add to $7i$:
$7i + (-7i) = 0$

-7i |  1    1+7i    7i
          ↓   -7i   -7i
        1      1      0

Answer:

The completed synthetic division table is:

-7i |  1    1+7i    7i
          ↓   -7i   -7i
        1      1      0

The resulting quotient polynomial is $x + 1$.