Question

Part V: If −1 is one of the roots of H(x) = x3 − 7x2 − x + 7, use synthetic division to find the resulting quadratic expression. Hint: If −1 is a root, (x + 1) is the factor. (2 points)
Part VI: Factor the quadratic expression from Part V to find the final 2 factors and roots. (2 points)

Answer 1

Solve for x over the real numbers:
x^3 - 7 x^2 - x + 7 = 0
The left hand side factors into a product with three terms:
(x - 7) (x - 1) (x + 1) = 0
Split into three equations:
x - 7 = 0 or x - 1 = 0 or x + 1 = 0
Add 7 to both sides:
x = 7 or x - 1 = 0 or x + 1 = 0
Add 1 to both sides:
x = 7 or x = 1 or x + 1 = 0
Subtract 1 from both sides:
Answer:  x = 7 or x = 1 or x = -1