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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

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.

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
#3
0

You are welcome. Jun 24, 2014