Polynomials Question (1 Viewer)

[phoenix159]

When P(x) is divided by (x - c), the remainder is c²
When P(x) is divided by (x - d), the remainder is d²
Find the remainder when P(x) is divided by (x - c)(x - d)

 

[HeroicPandas]

First one:

P(c) = c^2 ...(1)

Second one:

P(d) = d^2 ... (2)

P(x) = (x-c)(x-d)Q(x) + px + q (remainder has a degree less than of the divisor - it can be a consant or linear)

Apply ..(1)
P(c) = 0 + pc + q = c^2

pc + q = c^2 ..(3)

Apply ..(2)
P(d) = 0 + pd + q = d^2

pd + q = d^2 ..(4)

Simultaneously solve for p and q via equations (3) and (4)

Now (3) - (4)

p(c-d) = c^2 - d^2
p(c-d) = (c-d)(c+d) (difference of 2 ☐)

therefore, p = (c+d) (as {c-d} cannot be zero. If c-d = 0, then c = d, which is impossible as c and d are two distinct roots of polynomial P(x))

Sub this p into equation (3) or (4)

I'll sub into equation (3)

Therefore,

c(c+d) + q = c^2
c^2 + cd + q = c^2
q = -cd

Therefreo,

Since reminader is (px+q)

Therefore R(x) = (c+d)x - cd