# The polynomial of degree 5, P(x) has leading coefficient 1, has roots of multiplicity 2 at x=1 and x=0, and a root of multiplicity 1 at x=-3, how do you find a possible formula for P(x)?

##P(x) = x^5+x^4-5x^3+3x^2##

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The polynomial of degree 5, P(x) has leading coefficient 1, has roots of multiplicity 2 at x=1 and x=0, and a root of multiplicity 1 at x=-3, how do you find a possible formula for P(x)?
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Each root corresponds to a linear factor, so we can write:

##P(x) = x^2(x-1)^2(x+3)##

##=x^2(x^2-2x+1)(x+3)##

##= x^5+x^4-5x^3+3x^2##

Any polynomial with these zeros and at least these multiplicities will be a multiple (scalar or polynomial) of this ##P(x)##

Footnote

Strictly speaking, a value of ##x## that results in ##P(x) = 0## is called a root of ##P(x) = 0## or a zero of ##P(x)##. So the question should really have spoken about the zeros of ##P(x)## or about the roots of ##P(x) = 0##.

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