Find a polynomial of degree #3# that has roots #0# and #1#?

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Answer 1 · 2017-08-28T10:03:32

#P(x)=x^3-x# is one such solution

By the fundamental Theorem of Algebra, any polynomial of degree #3# can be written in the form:

# P(x) = A(x-alpha)(x-beta)(x-gamma) #

Where, #alpha,beta,gamma# are the roots (or zeros) of the equation #P(x)=0#

We are given that #0# and #1# are zeros of #P(x)=0#, thus, wlog, we can write:

# alpha = 0 #
# beta = 1 #

And so we have:

# P(x) = Ax(x-1)(x-gamma) #

We are free to choose any suitable #A#, or #gamma#, so let use choose:

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