Factor fx into linear factors given that k is a zero of fx 1

Factor f(x) into linear factors given that k is a zero of f(x)

1. f(x)= 3x^3 + 4x^2 - 69x + 90 when k=3

given k is a zero

first plug x=3 in f(x)

f(3) = 3(3)^3 +4(3)^2 -69(3) +90

=81 +36 -207+90

=207-207 =0

so \'3\' is a zero then we can write f(x) = (x-3) (ax^2 +bx+c)

3x^3 + 4x^2 - 69x + 90 = ax^3+bx^2+cx-3ax^2 -3bx-3c

3x^3 + 4x^2 - 69x + 90 = ax^3 +x^2(b-3a) +x(c-3b) -3c

comparing terms we get a=3 , -3c=90 c=-30

b-3a= 4

b-3(3) =4

b=4+9=13

so we can factor f(x) = (x-3) (3x^2 +13x -30)

now the roots of (3x^2 +13x -30) we have to find

roots are = [-13+ sqrt(13^3 -4(3)(-30) ] 2x3 and [-13- sqrt(13^3 -4(3)(-30) ] 2x3

= -13 +sqrt(169 +360) / 6 and -13 -sqrt(169 +360) / 6

=(-13+23)/6 and (-13-23)/6

= 10/6 and -36/6

= 5/3 and -6

f(x) =(x-3) (x-5/3) (x+6)