Question

If the cubic polynomials $x^3+a{x}^2+11x+6$ and $x^3+b{x}^2+14x+8$ may have a common factor of the form $x^2+px+q,$ then

• A. $a-p=b+q$
• B. $ap<bq$
• C. $pq$ divides $ab$
• D. $p+q$ divides $a+b$

Answer 1

Answer: (A)

$a-p=b+q$

Let, x³+ax²+11x+6=0 …(1)

And, x³+bx²+14x+8=0 …(2)

Subtract (1) from(2)

(b-a)+3x+2=0 …(3)

x²+px+q = 0 …(4)

Comparing (3) and(4)

We get, b-a=1 =>b=a+1 , p=3 and q=2

Now, b+q=(a+1)+2= a+3 =a+p

So, a+p=b+q