Question

PLEASE HELP, WILL FOLLOW AND MEDAL :)

Determine which consecutive integers do not have a real zero of f(x) = x^3+5x^2-x-6 between them?
the choice answers are:
- (-6,-5)
- (-5,-4)
- (-2,-1)
- (1,2)

Answer 1

I've been stuck on this since yesterday, I asked my math tutor but he couldn't get it too.

Answer 2

Ok lets solve this

Answer 3

Ok :)

Answer 4

first let me know what do you mean by root

Answer 5

I mean what you know about root?

Answer 6

well... its root and zero are the same thing

Answer 7

the*

Answer 8

yeah

Answer 9

that's pretty much it.. sorry

Answer 10

See root or zero is the value of x for which f(x)=0
i.e. where the curve cuts the x-axis, we get a root

Answer 11

okay, i see.

Answer 12

The given equation is a cubic equation and every cubic equation is continuous (in general every polynomial equation is continuous).

Answer 13

continuous means the curve will have no break..
Now we have the following theorem: If f(a).f(b)<0, then we have at least one zero of f in [a,b]

Answer 14

ok, so where do we start?

Answer 15

the roots of this equation are ±1,±2,±3,and±6 right?

Answer 16

here f(x)=x^3+5x^2-x-6
f(1)=-1 and f(2)=20 so there is a zero of the equation f(x)=0 in between 1 and 2 i.e. in the interval (1,2)

Answer 17

okay, nevermind what I said then lol

Answer 18

also f(-2)=8 and f(-1)=-1 so we have again a zero of the equation f(x)=0 in (-2,-1)

Answer 19

ok, so it can't be c.

Answer 20

also f(-5)=-1 and f(-4)=14 so a zero of f(x)=0 is in (-5,-4)

Answer 21

we already got 3 zeros, which is the maximum a cubic equation can have

Answer 22

so the answer is (-6,-5)

Answer 23

thanks!

Answer 24

Now do you want me to write the solution which you should write in exam or in assignment or you can do it yourself?

Answer 25

i can do it! :)

Answer 26

ok.:) You write the theorem
" If f is continuous and f(a).f(b)<0 then there is a zero of the equation f(x)=0 in (a,b)"

Answer 27

we did this problem based on the said theorem and also on another theorem which says
"an equation of degree n has n roots". we used this theorem in the part we said that since we got 3 roots, which is maximum, there can be no more roots and hence we discarded the interval (-6,-5)

Answer 28

@tvinsant

Answer 29

wow, you're really smart. I could never make something like that.