polynomials

Question

Given that x^2+px+q and 3x^2+q have a common factor x-b, where p,q and b are non-zero, show that 3p^2+4q=0.

Answer 1

By the Factor Theorem, if x-c is a factor of a polynomial f(x),

then f(c) = 0.

 

Since x-b is a factor of 3x²-q, 3b²-q = 0.  So, q = 3b²

 

Since x-b is a factor of x²+px+q, b²+pb+q = 0

 

Therefore, b²+pb+3b²=0

 

               4b²+pb=0

 

               b(4b+p)=0

 

               By assumption, b is nonzero.  So, p = -4b.

 

Consequently, 3p²+4q = 3(-4b)²+4(3b²) = -12b²+12b² = 0.