Text: Use synthetic division to show that g(x) is a factor of f(x), and complete the factorization of f(x).
g(x) = x - 4
f(x) = x^3 - 7x^2 - 28x + 160
To determine if g(x) is a factor of f(x), we can use synthetic division.
First, set up the synthetic division table:
4 | 1 -7 -28 160
Next, bring down the coefficient of the highest degree term, which is 1:
4 | 1 -7 -28 160
|___________
1
Multiply the divisor, 4, by the result, 1, and write the product below the next coefficient:
4 | 1 -7 -28 160
|___________
1
4
Add the product to the next coefficient:
4 | 1 -7 -28 160
|___________
1
4
-12
Repeat the process until all coefficients have been used:
4 | 1 -7 -28 160
|___________
1
4
-12
16
The last number in the bottom row represents the remainder. If the remainder is 0, then g(x) is a factor of f(x). In this case, the remainder is 16, so g(x) is not a factor of f(x).
To complete the factorization of f(x), we can use the quadratic formula or factor by grouping.