Question

Determine all factors of the expression 3x^3+7x^2-18x+8 if one of the factors is x-1

Answer 1

The factors are x-2,
`x-(2)/(3)` and x+4

Therefore the expression can be written as
`3x^3+7x^2-18x+8=(x-2)(x-(2)/(3))(x+4)`

Explanation:

Given expression is
`3x^3+7x^2-18x+8`

And also given that x-1 is one of the factors

i.e.,x-1=0

x=1

To find the factors equate the given expression to zero

`3x^3+7x^2-18x+8=0`

Using synthetic division to find the factors

1_| 3 7 -18 8

0 3 10 -8

___________________

3 10 -8 0

Therefore the quadratic equation is
`3x^2+10x-8=0`

To find the factors of the above equation:

`x=(-b\pm√(b^2-4ac))/(2a)`

Where a and b are coefficients of
`x^2` and x respectively

`x=(-10\pm√(10^2-4(3)(-8)))/(2(3))` Where a=3 , b=10 and c=-8

`=(-10\pm√(100+96))/(6)`

`=(-10\pm√(196))/(6)`

`=(-10\pm14)/(6)`

`x=(-10\pm14)/(6)`

Therefore
`x=(-10+14)/(6)` and
`x=(-10-14)/(6)`

`x=(4)/(6)` and
`x=(-24)/(6)`

Therefore
`x=(2)/(3)` and
`x=-4`

Therefore the factors are x-2,
`x-(2)/(3)` and x+4

`3x^3+7x^2-18x+8=(x-2)(x-(2)/(3))(x+4)`