two points and an equation

Question

So I think this is the question, I forget how its worded exactly, so bear with me. Suppose we have an equation $5x^3 + bx^2 + cx + d$ cirve that passes through the points $(0,0)$ and $(2,0)$ and the question asks me to solve for $b$ in the equation. How do I do it?

Answer 1

You don't have enough information for a unique solution

What you can do is:

$y = 5x^3 + bx^2+ cx + d$

Plug 0 in for x and 0 for y.

$d = 0$

Then plug 2 in for x and 0 for y

$40 + 4b + 2c = 0$

$b = -10 - \frac 12 c$

And that is as far as you can go.

Answer 2

For the point $(0, 0)$, you get

$$0 = 5(0)^3+b(0)^2+c(0)+d \iff d = 0$$

So the equation is essentially $f(x) = 5x^3+bx^2+cx$.

For the point $(2, 0)$, you get

$$0 = 5(2)^3+b(2)^2+c(2) \iff 0 = 40+4b+2c \iff -40 = 4b+2c \iff -20 = 2b+c$$

There isn’t enough information to solve for $b$. However, you can come up with a relation between $b$ and $c$, as shown.