How do you find all the zeros of #f(x)= x^3 - 2x^2 + 5x -10# with its multiplicities?

1 Answer
Aug 20, 2016

Answer:

#f(x)# has zeros #2# and #+-sqrt(5)i#

Explanation:

This cubic factors by grouping then using the difference of squares identity:

#a^2-b^2=(a-b)(a+b)#

with #a=x# and #b=sqrt(5)i# as follows:

#x^3-2x^2+5x-10#

#=(x^3-2x^2)+(5x-10)#

#=x^2(x-2)+5(x-2)#

#=(x^2+5)(x-2)#

#=(x^2-(sqrt(5)i)^2)(x-2)#

#=(x-sqrt(5)i)(x+sqrt(5)i)(x-2)#

Hence zeros:

#+-sqrt(5)i# and #2#