Question

Let $f\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$, where $f\left(x\right)g\left(x\right)q\left(x\right)$ and $r\left(x\right)$ are polynomials, and degree of $r\left(x\right)<deg.g\left(x\right)$. If deg $g\left(x\right)<deg.q\left(x\right)$ then deg. $g\left(x\right)$ is

• A. not defined
• B. 0
• C. deg. $f\left(x\right)-deg.g\left(x\right)$
• D. deg. $f\left(x\right)+deg.g\left(x\right)$

Answer 1

The correct option is A not defined

$Giventhatf\left(x\right)=g\left(x\right).q\left(x\right)+r\left(x\right)whendegr\left(x\right)<degg\left(x\right)Letustakedegreesoff\left(x\right)=a,g\left(x\right)=b,q\left(x\right)=qandr\left(x\right)=r.\therefore a=bq+rr<b----\left(1\right)Againgiventhatb<q----\left(2\right)from\left(1\right)b=\frac{a-r}{q}----\left(3\right)\left(2\right)b<q----\left(4\right)\left(3\right)and\left(4\right)cannotbesolvedsimultaneously.\therefore valueofbisnotdefined.OptionAiscorrect.$