Question

question 4 of 5
select the correct answer.
one factor of this polynomial is ((x - 7))(x^3 - 4x^2 - 22x + 7)
using synthetic division, what is the other factor of the polynomial?
(x^2 + 3x + 1)
(x^2 + 3x - 1)
(x^3 + 3x^2 - x)
(x^3 + 3x^2 + x)

Explanation:

Step1: Set up synthetic division

Use root $x=7$ (from factor $x-7$), coefficients of polynomial: $1, -4, -22, 7$

7 | 1  -4  -22   7
     ↓   7   21  -7
-------------------

Step2: Calculate column sums

Add values column-wise to get quotient coefficients.

7 | 1  -4  -22   7
     ↓   7   21  -7
-------------------
     1   3   -1   0

The quotient is $x^2 + 3x - 1$ (remainder 0 confirms divisibility).

Answer:

$\boldsymbol{x^2 + 3x - 1}$ (Option C)