Question on Factorising the Function

Topic : Factorization

Questions : g(x) =X^3-4^2-3X+18 , factor g(x) into three linear factors. By setting each factor equal to zero, find the three x-intercepts. What do you notice about two intercepts?

Solution :
f (x) = x³ – 4x² - 3x + 18

we can see if x=3;

f(3)=3³ – 4(3)² – 3(3) +18
=27 -36 -9 + 18
= 45 – 45
=0
f(3) = 0 , x =3 is a zero of f(x)
(x-3) is a factor

Now divide f(x) by (x-3)

x-3)x³ – 4x² – 3x +18 (x² – x – 6
-----x³ – 3x²
--------------------------
------- - x² – 3x
------- - x² + 3x
--------------------------
----------- - 6x +18
----------- - 6x + 18
--------------------------
------------------ 0

x² – x – 6 is the quotient, factorize it,
x² – x – 6 = (x-3)(x+2)
Hence f(x) = (x-3)(x-3)(x+2) = (x-3)² (x+2)