SOLUTION: If (x-2)^2 is a factor of x^3+px+q, find the value of p and q. Please help..

Algebra ->  Functions -> SOLUTION: If (x-2)^2 is a factor of x^3+px+q, find the value of p and q. Please help..       Log On


   

Question 824812: If (x-2)^2 is a factor of x^3+px+q, find the value of p and q. Please help..
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Let a be the other zero of the polynomial x%5E3%2Bpx%2Bq .
%28x-2%29%5E2%28x-a%29=x%5E3%2Bpx%2Bq
%28x%5E2-4x%2B4%29%28x-a%29=x%5E3%2Bpx%2Bq
x%28x%5E2-4x%2B4%29-a%28x%5E2-4x%2B4%29=x%5E3%2Bpx%2Bq
x%5E3-4x%5E2%2B4x-ax%5E2%2B4ax-4a=x%5E3%2Bpx%2Bq
x%5E3-4x%5E2-ax%5E2%2B4x%2B4ax-4a=x%5E3%2Bpx%2Bq
x%5E3-%284%2Ba%29x%5E2%2B%284%2B4a%29x-4a=x%5E3%2Bpx%2Bq
Since the polynomials above have the same value for all values of x ,
the coefficients of all terms must be the same:
1=1 for the term in x%5E3 ,
-%284%2Ba%29=0 for the term in x%5E2 ,
4%2B4a=p for the term in x , and
-4a=q for the independent term.

Solving for a :
-%284%2Ba%29=0
4%2Ba=0
highlight%28a=-4%29

Then,
p=4%2B4a
p=4%2B4%28-4%29
p=4-16
highlight%28p=-12%29
and
q=-4a
q=-4%28-4%29
highlight%28q=16%29