Question

____ is the polynomial which when divided by -x2+x-1 gives a quotient x-2 and remainder 3

Answer 1

Answer:

This states that there are polynomials qand r such that 

p(x)=q(x)(x−19)+91

 and p(x)=r(x)(x−91)+19. 

since (x−19)(x−91) is second-degree, the remainder must be first-degree, i.e. ax+b. 

p(x)=(x−19)(x−91)+ax+b

Since p(19)=q(19)(19−19)+91=91and 

p(19)=r(19)(19−19)+91=91, 

We have a system of equations

19a+b=91

91a+b=19

⇒b=91−19a and b=19−91a from above

⇒91−19a=19−91a⇒−72a=72

∴a=−1

Substitute a=−1 in 19a+b=91

⇒19×−1+b=91

∴b=91+19=110

Thus, we concluded that the remainder of p(x) by (x−19)(x−91) is equal to −x+110 or 110−x.