ANSWER TO QUESTION 1
Given
[tex]f(x)=5x^3+8x^2-7x-6[/tex]
We can use the factor theorem to determine if
[tex]x+2[/tex] is a factor of the polynomial or not.
According to this theorem, if [tex]x+2[/tex] is a factor of [tex]f(x)[/tex], then [tex]f(-2)=0[/tex].
How did we get the [tex]-2[/tex]?
We set [tex]x+2=0[/tex] and then solve to obtain [tex]x=-2[/tex].
So now let us plug in [tex]x=-2[/tex] in to the function to see if it will simplify to zero.
[tex]f(-2)=5(-2)^3+8(-2)^2-7(-2)-6[/tex]
[tex]f(-2)=5(-8)+8(4)+7(2)-6[/tex]
[tex]f(-2)=-40+32+14-6[/tex]
[tex]f(-2)=6-6[/tex]
[tex]f(-2)=0[/tex]
Since the result simplifies to zero, we conclude that
[tex]x+2[/tex] is a factor of
[tex]f(x)=5x^3+8x^2-7x-6[/tex]
ANSWER TO QUESTION 2
We have the function,
[tex]f(x)=5x^3+8x^2-7x-6[/tex]
We can use the remainder theorem to show that
[tex]x+1[/tex] is NOT a factor of the polynomial.
According to this theorem, if [tex]x+1[/tex] is not a factor of [tex]f(x)[/tex], then [tex]f(-1)\ne 0[/tex].
So now let us plug in [tex]x=-1[/tex] in to the function to see if it will simplify to non-zero number.
[tex]f(-1)=5(-1)^3+8(-1)^2-7(-1)-6[/tex]
[tex]f(-1)=5(-1)+8(1)+7(1)-6[/tex]
[tex]f(-1)=-5+8+7-6[/tex]
[tex]f(-1)=4[/tex]
[tex]f(-1)\ne0[/tex]
Since the result simplifies to a non zero number, we conclude that
[tex]x+1[/tex] is NOT a factor of
[tex]f(x)=5x^3+8x^2-7x-6[/tex]
