Complex Numbers And Quadratic Equations question 456
Question: Let $ a\ne 0 $ and p(x) be a polynomial of degree greater than 2. If p(x) leaves remainders a and -a when divided respectively, by x+a and x-a, the remainder when p(x) is divided by $ x^{2}-a^{2} $ is
Options:
A) 2x
B) -2X
C) x
D) -x
Show Answer
Answer:
Correct Answer: D
Solution:
[d] We are given that $ p(-a)=aandp(a)=-a $ [When a polynomial f(x) is divided by x-a, remainder is f(a)], Let the remainder, when p(x) is divided by $ x^{2}-a^{2} $ , be Ax+B. Then,. $ p(x)=Q(x)(x^{2}-a^{2})+Ax+B $ (1) Where Q(x) is the quotient. Putting x=a and -a in (1), we get $ p(a)=0+Aa+B\Rightarrow -a=Aa+B $ (2) And $ p(-a)=0-aA+B\Rightarrow a=-aA+B $ (3) Solving (2) and (3), we get B=0 and A=-1 Hence, the required remainder is-x.
BETA
