How do you find the zeros of #f(x) = x^3 + 4x^2 - 25x - 100#?

2 Answers
Lotusbluete
Mar 23, 2016

There are three zeros:
#x = -5 " or " x = 5 " or " x = -4#.

Explanation:

You should try and recognize patterns in your expression that would help you factorize.

For example, here, you can notice that #25# and #100# are both dividable by #25#, so you could try to to factor #25# and find the following factorization:

# x^3 + 4x^2 - 25x - 100 = x^2(x+4) - 25(x+4) = (x^2 - 25)(x+4) #

Now you can use the identity #a^2 - b^2 = (a+b)(a-b)# to factorize further:

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

Now,

#f(x) = 0#

#<=> x^3 + 4x^2 - 25x - 100 = 0#

#<=> (x+5)(x-5)(x+4) = 0#

A product is equal to zero if one or more factors is/are equal to zero:

#<=> x+5 = 0 " or " x-5 = 0 " or " x+4 = 0#

#<=> x = -5 " or " x = 5 " or " x = -4#

mason m
Mar 23, 2016

#{(x=-5),(x=5),(x=-4):}#

Explanation:

Factor by grouping:

#x^3+4x^2-25x-100#

#=(x^3+4x^2)-(25x+100)#

#=x^2(x+4)-25(x+4)#

#=(x^2-25)(x+4)#

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

This gives the solutions

#{(x=-5),(x=5),(x=-4):}#

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