if (x+2) is a factor of f(x) =x^3+3x^2-kx+4
what is k? I need this explained this is an example ,,, i dont understand how to solve this type of problems ! Your help is appreciated
if (x+2) is a factor of f(x) =x^3+3x^2-kx+4
then
(x+2)(something)=x^3+3x^2-kx+4
consider if x=-2 then
(-2+2)(something)=x^3+3x^2-kx+4
But -2+2=0 and if you multiply anything by zero then the answer is zero, that is;
(-2+2)(something)=x^3+3x^2-kx+4=0
So what this is saying is that (x+2) is a factor IF x=-2 is a root, that is, f(-2) =0
so
$$f(x)=x^3+3x^2-kx+4\\\\
f(-2)=(-2)^3+3(-2)^2-k(-2)+4\\\\
f(-2)=-8+12+2k+4\\\\
f(-2)=8+2k\\\\$$
Now if (x+2) is a factor then f(-2)=0 so
$$8+2k=0\\
2k=\:-8\\
k=\:-4\\$$
BY THE WAY this is called remainder theorum
If (x+a) is a factor of f(x) then f(-a)=0
I've just explained why. If you have a good maths brain this will help a lot.
if (x+2) is a factor of f(x) =x^3+3x^2-kx+4
then
(x+2)(something)=x^3+3x^2-kx+4
consider if x=-2 then
(-2+2)(something)=x^3+3x^2-kx+4
But -2+2=0 and if you multiply anything by zero then the answer is zero, that is;
(-2+2)(something)=x^3+3x^2-kx+4=0
So what this is saying is that (x+2) is a factor IF x=-2 is a root, that is, f(-2) =0
so
$$f(x)=x^3+3x^2-kx+4\\\\
f(-2)=(-2)^3+3(-2)^2-k(-2)+4\\\\
f(-2)=-8+12+2k+4\\\\
f(-2)=8+2k\\\\$$
Now if (x+2) is a factor then f(-2)=0 so
$$8+2k=0\\
2k=\:-8\\
k=\:-4\\$$
BY THE WAY this is called remainder theorum
If (x+a) is a factor of f(x) then f(-a)=0
I've just explained why. If you have a good maths brain this will help a lot.