Determining Quadratic and Cubic Functions with Integral Coefficients

[Dough]

Well there are actually two questions i need some help with, if you could provide a full solution so i cna see the step by stpe thing it would be nice.

Question 1
A quadratic function f(x) with integral coefficients has the following properties:
f(3/2) = 0, (x-2) is a factor of f(x), and f(4) = 50. Determine f(x).

Question 2
A cubic function g(x) with integral coefficients has the following properties:
g(3) = 0, G(-3/4) = 0, (x+2) is a factor of g(x), g(1) = -84. Determine g(x).

Thanks!

 

[EnumaElish]

You could begin by writing out a general (generic) quadratic function, e.g. y = ax² + bx + c and then think how you could solve for a, b and c. Does that help to get you started?

 

[Dough]

i've written out what I've been given:

f(3/2) = 0 therefore (2x-3) is a factor of f(x),

Factors of f(x) = (2x-3) and (x-2)
f(4) = 50

i also wrote out ax^2 + bx + c as oyu suggested but not seeing a link... :(

 

[stunner5000pt]

whati s the general form of f(x) ??
quadratics look like $f(x) = ax^2 + bx + c$
ok from taht what is f(2) ?? Using the formula for f(x) above and your given info form equations using f(3/2) and f(4) as well to solve for a, b, and c.

 

[EnumaElish]

Dough, follow stunner's suggestion. For example, how would you write f(4) in terms of $f(x) = ax^2 + bx + c$?

 

[Dough]

i did that but no luck i'll give it another shot, i make mistakes osmetimes so its possible i may have done somethign wrong whihc messed everythign else up...

 

[EnumaElish]

Go ahead and post here what you've done.

 

[Dough]

yay, i got the first one... on the third try after the above post, i kept makign silly errors just now... i'll let you know how i do witht he the next one, thanks for the help :D

 

[Dough]

i don't think i got it right and it was hell of a lot of work... i might give it a shot tomorrow or later i got f(x) = -8x^3 - 50x^2 - 54x + 28

thanks tho i got one so i have the idea now its just not makign stupid mistakes!

dry, we didnt even have to do number 3... i sitll try it agian later caus ei wanan find out how to do it!

 

[HallsofIvy]

By the way, Dough, I think you were doing the first problem in a better way than writing "f(x)= ax²+ bx+ c" as others were telling you. You are correct that since 3/2 is a 0 of f, 2x-3 is a factor. And you are told that x-2 is a factor. Since f is quadratic, the can't be any more factors involving x, only a constant:
f(x)= a(x-2)(2x-3). Now put x=4 in that, set it equal to 50 and solve for a.

Pretty much the same thing with question 2:
You know that 3 and -3/4 are zeroes of g (I presume that "G(-3/4)" was really "g(-3/4)" ) so that x-3 and 4x+3 are factors. You are told that x+ 2 is a factor. Those three factors will give a cubic so there are no other factors with an x. We must have
g(x)= a(x-3)(4x+3)(x+2) for some constant a. Since g(1) = -84, set x= 1, put it equal to -84 and solve for a.