Example 2. Use synthetic division to show that 5 is a root of p(x) = 2x³ - 40x - 50 = 0, and use this fact to solve the equation.

Solution: We divide p(x) by x - 5 with the object of showing that the remainder p(5) is zero. Thus:

This shows us that p(x) = (x - 5)(2x² + 10x + 10). The roots of p(x) = 0 are therefore obtained from

as 5 and we have, then

5,  and 

Example 3. Let f(x) = 9x³ + x² - 7x + 4. Find numbers a, b, c, and d such that
 

 
We give two solutions.

First solution: Letting x = -1 in (3), we see that a = f(-1). We therefore use synthetic division to express f(x) in the form (x + 1)g(x) + f(-1) and find that g(x) = 9x² - 8x + 1 and a = f(-1) = 3. Now (3) becomes
 

On each side we subtract 3 and then divide by x + 1, thus obtaining

 
Letting x = -1, we see that b = g(-1). We therefore treat g(x) as f(x) was treated above, and find that g(x) = (x + 1)(9x - 17) + 18. Hence b = 18. Then (4) becomes
 

This leads to