Question

If the roots of the equation x3 ax+ bx - c = 0 are three consecutive integers, then what is the smallest possible value of b?​

Answer 1

Answer:

Smallest possible value is −1

Step-by-step explanation:

Let's denote our polynomial as:

px^3+rx+qx+s=0

and x1, x2 and x3 will be the roots of that polynomial.

then:

(x1+x2+x3)×(x3+x2+x3)=c/a

Now back to our question:

our p=1, q=b

Roots of our polynomial are consecutive integers =(y−1), y and (y+1)

and we have:

y∗(y−1)+y∗(y+1)+(y−1)∗(y+1)=b

simplifying this we'll get:

3∗y2−1=b and we need to find min of that function.

Because y2 is always >=0, them our b will get min value only when y=0

Hence bmin= −1

Hope it was helpful for you