SOLUTION:
Step 1 :
In this question, we are given that x = - 1 is a zero of the polynomial:
[tex]g(x)=x^3+7x^2\text{ + 12 x + 6}[/tex]
If x = - 1 is a zero of the polynomial, then x + 1 is a factor of :
[tex]g(x)=x^3+7x^2\text{ + 7 x + 6}[/tex]
Step 2 :
[tex]\begin{gathered} \text{Then, we have that:} \\ \frac{x^3+7x^2\text{ + 12 x + 6}}{x\text{ + 1}} \\ =x^2\text{ + 6 x + 6} \end{gathered}[/tex]
Step 3 :
Expressing g( x ) as a product of linear factors, we have that:
[tex]\begin{gathered} \text{If g ( x ) = }x^3+7x^2\text{ +12 x + 6,} \\ \text{then g ( x ) = ( x + 1 ) ( x}^2\text{ + 6 x + 6 )} \end{gathered}[/tex]
CONCLUSION:
Expressing g ( x ) as a product of linear factors, we have that :
[tex]g(x)=(x+1)(x^2\text{ + 6x + 6 )}[/tex]
