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- Homework Statement
- If ##4x^3+kx^2+px+2## is divisible by ##x^2+λ^2##. Prove that ##kp=8##

- Relevant Equations
- Factor theorem

My attempt;

##4x^3+kx^2+px+2=(x^2+λ^2)(4x+b)##

##4x^3+kx^2+px+2=4x^3+bx^2+4λ^2x+bλ^2##

##⇒k=b, p=4λ^2 , bλ^2=2##

##\dfrac{4λ^2}{bλ^2}=\dfrac{p}{2}##

##\dfrac{4}{b}=\dfrac{p}{2}##

##⇒8=pb## but ##b=k##

##⇒8=kp##

Any other approach appreciated...

##4x^3+kx^2+px+2=(x^2+λ^2)(4x+b)##

##4x^3+kx^2+px+2=4x^3+bx^2+4λ^2x+bλ^2##

##⇒k=b, p=4λ^2 , bλ^2=2##

##\dfrac{4λ^2}{bλ^2}=\dfrac{p}{2}##

##\dfrac{4}{b}=\dfrac{p}{2}##

##⇒8=pb## but ##b=k##

##⇒8=kp##

Any other approach appreciated...