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

Guest Jun 24, 2014

#1**+10 **

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.

Melody
Jun 24, 2014

#1**+10 **

Best Answer

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.

Melody
Jun 24, 2014