Answer
$f(x)=5x^{3}-10x^{2}+5x$
Work Step by Step
If $k$ is a zero, then $(x-k)$ is a factor of $f(x)$ ... (factor theorem)
So,
$(x-0)=x$ is a factor of f,
$(x-1)$ is a factor of f.
The number of times $(x-k)$ occurs as a factor is referred to as the multiplicity of the zero.So,
$x$ is a factor of f, degree=1
$(x-1)^{2}$ is a factor of f, degree = 2
(the sum of degrees is 3)
Since f has degree 3, $\quad f(x)=ax(x-1)^{2}$
To find $a,$ use the given information: $f(2)=10$
$a(2)(1)^{2}=10$
$2a=10$
$a=5$
Thus,
$f(x)=5x(x-1)^{2}$
$=5x(x^{2}-2x+1)$
$f(x)=5x^{3}-10x^{2}+5x$
