# Thread: Zeros of a cubic function

1. ## Zeros of a cubic function

Hello,

I'm trying to help my brother with a math problem, but I just haven't done this stuff in quite a while so I don't even know where to start. I'd appreciate any assistance!

Cheers.

Find the zeros for the function f(x) = ax^3 + bx^2 -106x + d, given that f(-1) = 343, f(2) = 40, and f(3) = -33.

2. So I started an attempt at this by solving for the given values, then reducing two pairs of equations down to two equations with two unknowns (I eliminated d). Then I substituted b as a function of a from one equation into the other, but I got a = -(8/14), which leads me to believe I made a mistake (this is a high school question, so I expect numbers to work out nicely).

3. Originally Posted by RandomGuy8181
Hello,

I'm trying to help my brother with a math problem, but I just haven't done this stuff in quite a while so I don't even know where to start. I'd appreciate any assistance!

Cheers.

Find the zeros for the function f(x) = ax^3 + bx^2 -106x + d, given that f(-1) = 343, f(2) = 40, and f(3) = -33.
$
f(x) = ax^3+bx^2-106x+d
$

Use, f(-1) = 343, we got

$343 = a(-1)^3+b(-1)^2-106(-1) +d$

$\Rightarrow 343= -a+b+106+d$

$\Rightarrow -a+b+d=237$ ..................(1)

Now, Use, f(2) = 40, we got

$40 = a(2)^3+b(2)^2-106(2) +d$

$\Rightarrow 343= 8a+4b-212+d$

$\Rightarrow 8a+4b+d=555$ ..................(2)

By Using, f(3) = -33, we got

$33 = a(3)^3+b(3)^2-106(3) +d$

$\Rightarrow 27a+9b+d=351$ ..................(3)

Now, subtract eqn (1) from eqn(2) and eqn(3),
that means, (2) - (1) and (3) - (1), we got

9a + 3b = 318 .........................(4)

28a + 8b = 114 .......................(5)

divide eqn(4) by 3 and eqn (5) by 2

3a + b = 106

14a + 4b = 57

Now solve these two eqns and find a and b. then put those values of a and b in eqn (1) to find d. Then you will have a cubic function.
Now, after that use factor theorem to find one factor. Then divide the cubic function with that factor to find a quadratic factor. Then factor that quadratic function. Finish it.