Question

Find a cubic function with the given zeros.

√6 , -√6, -3

Answer 1

@amistre64

Answer 2

what do we know about zeros of a function, what do they do to the function?

Answer 3

they mark the intercepts?

Answer 4

yeah, they do mark the x intercepts.

but something more useful to is is that the make the function, equal out to zero.

Answer 5

now, another thing we should know, is that anything times 0, is equal to 0 ... right?

Answer 6

yep

Answer 7

so, would you agree that if x=a, then x-a = 0 ?

Answer 8

yes

Answer 9

given any number of roots, we can then create a function:

say the roots are a,b,c

let f(x) = (x-a)(x-b)(x-c)

will f(a)=0? f(b)=0? and f(c)=0?

Answer 10

okay, so we have to find which one ?

Answer 11

we simply need to expand (multiply thru, distribute) this into the cubic format.

Answer 12

let f(x) = (x-a)(x-b)(x-c) for the stated zeros; a,b, and c

how would we expand this into a cubic format?

Answer 13

multiply it out

Answer 14

can you show me how it looks when we multiply it out?

Answer 15

http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427elr1dfihajj is that right?

Answer 16

that link is not working for my browser :/

(x-c)(xx-bx-ax+ba)

xxx-bxx-axx+bax -cxx+bcx+cax-abc

x^3 -(a+b+c)xx^2 +(ab+ac+bc)x -abc

Answer 17

let a=sqrt(6), b=-sqrt(6), and c=-3

Answer 18

ok

Answer 19

btw the link says x^3 + 3x^2 -6x -18

Answer 20

a+b+c = -3, so --3 = 3 is fine

ab+ac+bc = -6, thats good

and -(--6.3) is -18

Answer 21

so now we just combine them to get the function

Answer 22

that IS the function ..

Answer 23

f(x) = x^3 +3x^2-6x-18

Answer 24

any scaled form of this also works ... so this is not a unique solution

Answer 25

i see, thanks so much!

Answer 26

goodluck