How do you write a polynomial function given the real zeroes 1,-4, 5 and coefficient 1?

2 Answers
George C.
Dec 6, 2015

The simplest such polynomial is:

#f(x) = (x-1)(x+4)(x-5) = x^3-2x^2-19x+20#

Explanation:

Convert the zeros into linear factors to find:

#f(x) = (x-1)(x+4)(x-5) = x^3-2x^2-19x+20#

Any polynomial in #x# with these zeros will be a multiple (scalar or polynomial) of this #f(x)#.

sente
Dec 6, 2015

Begin from the factored form to find the desired polynomial to be
#f(x) = x^3 -2x^2 - 19x + 20#

Explanation:

An easy way of generating a polynomial with a given set of zeros is to begin with the factored form. A polynomial #f(x)# has #a# as a zero if and only if #(x-a)# is a factor of #f(x)#.

Using this, we can construct the desired polynomial as follows:

#f(x) = (x-1)(x-(-4))(x-5)#

#=(x-1)(x+4)(x-5)#

#=x^3 -2x^2 - 19x + 20#

Note that we could multiply by a constant to give #x^3# a different coefficient, however this method naturally produces the coefficient of the highest power as #1#.

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