# The polynomial of degree 5, P(x) has a leading coefficient 1, has roots of multiplicity 2 at x=2 and x=0, and a root of multiplicity 1 at x=-4 and it goes through the point (5, 18) how do you find a formula for p(x)?

May 26, 2016

Such a polynomial does not exist

#### Explanation:

The conditions of leading coefficient 1, roots multiplicity 2 at $x = 2$ and $x = 0$ and a single root at $x = - 4$ are enough to define uniquely the polynomial
${p}_{5} \left(x\right) = {\left(x - 2\right)}^{2} {x}^{2} \left(x + 4\right)$
but this polynomial verifies ${p}_{5} \left(5\right) = 2025 \ne 18$