Write a polynomial equation Write a polynomial equation with the following characteristics. 
A quartic function with roots of -3, -1, and 4 (x=4 has a multiplicity of 2) and which passes through the point (5,16)
I know how to relate the roots to a polynomial equation, but I don't know how the point (5,16) is related to the function, so I cannot complete the equation. Any help would be appreciated. 
 A: Let 
$$ f(x) = \alpha(x+3)(x+1)(x-4)^2 $$
Then solve $f(5) = 16$ for $\alpha$. Plug that $\alpha$ into $f(x)$ as given above, and you're done.
Here I have used the general form
$$f(x) = \alpha(x-r_1)(x-r_2)(x-r_3)(x-r_4)$$
where $\alpha$ is the coefficient of $x^4$, and $r_i$ are roots.
A: with the ansatz
$f(x)=ax^4+bx^3+cx^2+dx+e$ we have
$f(-3)=0$
$f(1)=0$
$f(4)=0$
$f'(4)=0$
and $f(5)=16$
you will get an equation system and you must solve this.
A: Well generally your equation will look like $(x-\alpha)(x-\beta)(x-\delta)(x-\gamma)=y$ where $\alpha,\beta,\delta,\gamma$ are your roots. Multiplying the entire equation by a constant, say $k$ will not change your root values. Try creating your function and solving for $k$ using your point on the line.
I should state that $(x-\alpha)(x-\beta)(x-\delta)(x-\gamma)=0$ is the same as $k(x-\alpha)(x-\beta)(x-\delta)(x-\gamma)=0$, since you can clearly divide both sides by k and get back the first equation. Thus the roots should remain intact.
A: As you already figured out, for your particular problem you can write a polynomial
with the required zeros like this, for suitable $a,b,c$:
$$f(x) = (x+a)(x+b)(x+c)^2.$$
And as has been noted, this polynomial also has the same roots:
$$g(x) = k(x+a)(x+b)(x+c)^2.$$
So write $f(x)$ in whatever format you need, and evaluate $n = f(5).$
If $n = 16,$ you would be done.
But if not, you at least want $g(5) = 16.$
Now you have $f(5)=n$ for a known value $n,$ and $g(5) = kf(5) = 16.$
This should be sufficient to find $k$ and to write $g(x)$ in the desired format.
