Answer:
Option B - \(f(x)=3x^3+12x^2+3x-18=3(x-1)(x+2)(x+3)\)
Step-by-step explanation:
Given : Polynomial \(f(x)=3x^3+12x^2+3x-18\)
To find : Express the polynomial as a product of linear factors?
Solution :
Factor of the polynomial \(f(x)=3x^3+12x^2+3x-18\)
Taking 3 common
\(f(x)=3(x^3+4x^2+x-6)\)
Now, We factor the cubic term by rational root theorem.
If a polynomial function has integer coefficients, then every rational zero will have the form \(\frac{p}{q}\) where p is a factor of the constant and q is a factor of the leading coefficient.
\(p=\pm(1,2,3,6)\\q=\pm1\)
The possible roots of the polynomial function is
\(\pm1,\pm2,\pm3,\pm6\)
Now, we substitute the values in the polynomial if it is equal to zero then it is the root.
Substitute x=1
\(f(1)=1^3+4(1)^2+1-6=0\)
So, x=1 is one of the root.
Similarly we substitute all the values one by one.
The values satisfied is x=-2 and x=-3
Substitute x=-2
\(f(-2)=(-2)^3+4(-2)^2-2-6=0\)
So, x=-2 is one of the root.
Substitute x=-3
\(f(-3)=(-3)^3+4(-3)^2-3-6=0\)
So, x=-3 is one of the root.
So, The factors of \(f(x)=x^3+4x^2+x-6\) is (x-1)(x+2)(x+3).
Therefore, The linear factor of the given polynomial is
\(f(x)=3x^3+12x^2+3x-18=3(x-1)(x+2)(x+3)\)
So, Option B is correct.
