How do you factor #x^3 - 9x^2 + 24x - 20#?

1 Answer
Aug 18, 2016

Answer:

#x^3-9x^2+24x-20 = (x-2)(x-2)(x-5)#

Explanation:

#f(x) = x^3-9x^2+24x-20#

By the rational root theorem, any rational zeros of #f(x)# are expressible in the form #p/q# for integers #p, q# with #p# a divisor of the constant term #20# and #q# a divisor of the coefficient #1# of the leading term.

That means that the only possible rational zeros are:

#+-1, +-2, +-4, +-5, +-10, +-20#

Trying each in turn, we find:

#f(2) = 8-9(4)+24(2)-20 = 8-36+48-20 = 0#

So #x=2# is a zero and #(x-2)# a factor:

#x^3-9x^2+24x-20 = (x-2)(x^2-7x+10)#

Note that #7 = 2+5# and #10 = 2*5#

So we find:

#x^2-7x+10 = (x-2)(x-5)#

Putting it all together:

#x^3-9x^2+24x-20 = (x-2)(x-2)(x-5)#