Question

if - px+qx+6+x3has (x-1) as a factor and leaves a remainder 4 when divided by (x+1) find the values of p and q​

Answer 1

Answer:

P(x) when divided by x−3 and x−5 leaves remainder 10 and 6 respectively.

From polynomial-remainder theorem, P(3)=10 and P(5)=6

If the polynomial is divided by (x−3)(x−5) then remainder must be of the form ax+b (degree of remainder is less than that of divisor)

⇒P(x)=Q(x)(x−3)(x−5)+(ax+b), where Q(x) is some polynomial.

Substituting for x=3 and x=5:

P(3)=10=3a+b

P(5)=6=5a+b

Solving for a and b, we get

a=−2 and b=16

⇒Remainder=−2x+16