How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 0, 2, 2, 4?

${x}^{4} - 8 {x}^{3} + 20 {x}^{2} - 16 x = 0$

Explanation:

$x = 0 , x = 2 , x = 2 , x = 4$

These are called "zeros" because we're looking for where the graph crosses the X-axis - or in other words, where "y=0". So we'll have:

$x = 0 , x - 2 = 0 , x - 2 = 0 , x - 4 = 0$

We do this because in a function, if one term within an equation that is all multiplication equals 0, the entire function equals 0. So we get:

$x \left(x - 2\right) \left(x - 2\right) \left(x - 4\right) = 0$

And now it's just expanding all these terms into one polynomial equation:

$x \left({x}^{2} - 4 x + 4\right) \left(x - 4\right) = 0$

$x \left({x}^{3} - 8 {x}^{2} + 20 x - 16\right) = 0$

${x}^{4} - 8 {x}^{3} + 20 {x}^{2} - 16 x = 0$