How do you determine whether x+2 is a factor of the polynomial #2x^3+2x^2-x-6#?

3 Answers
May 31, 2018

You divide using polynomial long division or synthetic division and if the remainder is 0 then it is a factor.

Explanation:

#2x^3+2x^2-x-6 divide x+2#

this division has a remainder of -12 so it is not a factor.

May 31, 2018

Just check if #x=-2# is a root of polynomial. See below

Explanation:

We know that if a polynomial #P(x)# has a root for #x=a#, then #P(a)=0# and #x-a# divides to #P(x)# (this is the same that #x-a# is a factor of #P(x)#

Lets see.

#P(-2)=2·(-2)^3+2·(-2)^2-(-2)-6=-16+8+2-6=-12!=0#

Then #x+2# is not a factor of #P(x)#

May 31, 2018

#x+2# not a factor of #f(x)#

Explanation:

Let #f(x)=2x^3+2x^2-x-6#

We can use the Polynomial Remainder Theorem, which states when a polynomial, #f(x)# is divided by #x-c#, the remainder is #f(c)#.

We are dividing #f(x)# by #x+2#, so #c=-2#. Now, let's input this into #f(x)#. We get

#2(-2)^3+2(-2)^2-(-2)-6#

#=>2(-8)+2(4)+2-6#

#=>-16+8+2-6#

#=>color(blue)(-12)#

We have a remainder, which means #x+2# is not a factor of #f(x)#.

If #f(c)# simplified to #0#, we would have no remainder, and #x+2# would be a factor, but since we have a remainder, #x+2# is not a factor.

Hope this helps!