What are all the possible rational zeros for #f(x)=x^3+x^2-5x+3# and how do you find all zeros?

1 Answer
Nov 23, 2016

1,1, -3


Substituting x=1 makes f(x)= 1+1-5+3 =0 hence x=1 is one of the rational zeros.

Since x=1 is a zero, x-1 would be a factor of f(x). therefore divide f(x) by x-1 by using long or synthetic division.
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So, #f(x)= (x-1)(x^2+2x-3)#

Now for finding other zeros of f(x), put #x^2+2x-3=0# and solve for x.

This quadratic equation can be solved by factorisation or using quadratic formula.

Factorisation is quite easy. Writing 2x as 3x-x, the quadratic expression becomes #x^2 +3x -x-3=0#

#x(x+3) -1(x+3)=0#

(x-1)(x+3)=0 giving x= 1, -3

Thus all the zeros of f(x) are now known. These are, 1( which is a repeated zero) and -3