find the values of p and q if (x-1)^2 is a factor of x^3+4x^2-(2p-q)x+pq

Question

find the values of p and q if (x-1)^2 is a factor of x^3+4x^2-(2p-q)x+pq

Answer 1

You can do this via long division, dividing x³ + 4x² - (2p-q)x + pq by x² -2x + 1 (the same as (x-1)²).  In order for (x-1)² to be a factor of x³ + 4x² - (2p-q)x + pq, the latter would have to be exactly divisible by the former, with a remainder of 0.

 

When you work out the division, the remainder you get is (q - 2p + 11)x + (pq - 6).  For this to be 0, two things have to be true:

 

1. q - 2p + 11 = 0

2. pq - 6 = 0

 

This is a system of two equations with two unknowns that can be solved:

 

q - 2p + 11 = 0

q = 2p - 11

 

pq - 6 = 0

p(2p - 11) - 6 = 0

2p² - 11p - 6 = 0

 

This can solved by factoring:

(p - 6)(2p + 1) = 0

 

So p = 6, or p = -1/2

 

If p = 6, q = 1

If p = -1/2, q = -12

 

So there are two solutions:  p=6 and q=1, or p=-1/2, q=-12

 

In both cases, the coefficient of the x in the original equation is 11, and the constant term is 6.