Question 997336

If you know that -2 is a zero of f(x)=x^3+7x^2+4x-12, explain how to solve the equation x^3+7x^2+4x+12=0. 
Also give an explanation to why x-2 is or is not a factor
<pre>{{{f(x) = x^3 + 7x^2 + 4x - 12}}}

Itw's given that - 2 is a zero

Using synthetic division, we get:

  - 2| 1    7    4  - 12 
     |_____<b><u>-2 - 10    12</b></u>  
       1    5  - 6     0

Since - 2 is a zero, then x = - 2, and x + 2 = 0, so x + 2 is a factor

We now get: {{{f(x) = (x + 2)(x^2 + 5x - 6)}}}

{{{f(x) = (x + 2)(x + 6)(x - 1)}}} ------- Factoring {{{x^2 + 5x - 6}}}

{{{0 = (x + 2)(x + 6)(x - 1)}}}

Therefore, {{{highlight_green(system(x = - 2_OR,x = - 6_OR,x = 1))}}}


If x - 2 is a factor, then x - 2 = 0, and so, x = 2 (one of the zeroes)

If x - 2 is a factor, then 2 (one of the zeroes) should NOT produce a remainder when the Remainder theorem: is used.

Remainder theorem: {{{f(x) = x^3 + 7x^2 + 4x - 12}}}

Therefore, we get: {{{f(2) = 2^3 + 7(2)^2 + 4(2) - 12}}}
 
{{{f(2) = 8 + 28 + 8 - 12}}}

{{{f(2) = 32}}}

With f(2) = 32, a remainder exists, and so, x - 2 is NOT a factor</pre>