Roots Due to Number Theory.

Algebra Level 3

$\large x^3 - x^2 - 5x + 1 = 0$

If $\alpha,\beta$ and $\gamma$ are the roots of the equation above, find the value of $\lfloor \alpha \rfloor + \lfloor \beta \rfloor + \lfloor \gamma \rfloor$ .

The answer is 0.

1 solution

Akshat Sharda
Jan 31, 2018

Let $x^3-x^2-5x+1=f(x)$ ,

By observation,

• $f(-2)<0$ and $f(-1)>0$

• $f(0)>0$ and $f(1)<0$

• $f(2)<0$ and $f(3)>0$

So, one root each lie in $(-2,-1),(0,1)$ and $(2,3)$ .

Therefore, the floor function of the roots will give us $-2,0$ and $2$ . So, $-2+0+2=0$ .

@Akshat Sharda, but what prompted you to check for -2, -1, 0, 1 and 2 only? Why you thought of these numbers only?

Priyanshu Mishra - 3 years, 4 months ago

What is "differential theory of function"?

Pi Han Goh - 3 years, 4 months ago

Nothing. Just to check f’(x) And higher derivatives and checking concavity and convexity.

Priyanshu Mishra - 3 years, 4 months ago

@Priyanshu Mishra – I don't see how that is helpful here. Can you elaborate more on this?

Pi Han Goh - 3 years, 4 months ago

@Pi Han Goh – @Pi Han Goh , I basically want to know why he checked the value of function at that points only? Why not other points? That’s my problem only.

Priyanshu Mishra - 3 years, 4 months ago

@Priyanshu Mishra – Please read up Intermediate value theorem .

Pi Han Goh - 3 years, 4 months ago

Roots Due to Number Theory.

The answer is 0.

1 solution

0 pending reports