Question

P(x) be any polynomial. When it is divided by $(x-17)$ and $(x-71)$ , then the remainders are 71 and 17 respectively. Find remainder, when $P\left(x\right)$ is divided by
$(x-17)(x-71)$.

• A.
  $x-88$
• B.
  $88$
• C.
  $88-x$
• D.
  $88x+11$

Answer 1

The correct option is C

$88-x$
Given :

$P\left(x\right)=(x-17)A\left(x\right)+71\Rightarrow P\left(17\right)=71P\left(x\right)=(x-71)B\left(x\right)+17\Rightarrow P\left(71\right)=17P\left(x\right)=(x-17)(x-71)k\left(x\right)+(ax+b)(A-B)+54x+(17-71)(17+71)\Rightarrow P\left(17\right)=0+17a+b\Rightarrow 71=17a+b......\left(1\right)\Rightarrow P\left(71\right)=0+71a+b\Rightarrow 17=71a+b......\left(2\right)Now,subtract\left(1\right)from\left(2\right)\Rightarrow -54=54a\Rightarrow a=-1\Rightarrow 71=-17+b\Rightarrow b=88So,theRemainderis(-x+88)Hence,optionciscorrect.$