Given polynomial #f(x)=x^3-10x^2+19x+30# and a factor #x-6# how do you find all other factors?

1 Answer
sjc
Jun 9, 2017

#:.x^3-10x^2+19x+30=(x-6)(x-5)(x+1)#

Explanation:

if #(x-6)# is a factor we have

#x^3-10x^2+19x+30=(x-6)(x^2+bx+c)#

#color(white)(xxxxxxxxxxxxxxx)=x^3+bx^2+cx#

#color(white)(xxxxxxxxxxxxxxx)=color(white)(xx)-6x^2-6bx-6c#

#color(white)(xxxxxxxxxxxxxxx)=x^3+color(blue)((b-6))x^2+color(red)((c-6b))x-6c#

we now compare coefficients and solve

(Coefficients of #x^2)#

#LHS=-10#

#RHS=b-6#

#color(blue)(b-6=-10=>b=-10+6)#

#b=-4#

Coefficients of x

#LHS=19#

#RHS=c-6b#

#color(red)(-6xx-4+c=19)#

#=>24+c=19#

#c=-5#

check constant term

#LHS=30#

#RHS=-6 xx -5=30#

#:.x^3-10x^2+19x+30=(x-6)(x^2-4x-5)#

now see if the quadratic factorises

factors #-5# that sum to #-4#

#-5" "#&#" "=+1#

we have therefore

#:.x^3-10x^2+19x+30=(x-6)(x-5)(x+1)#

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