Polynomials-types, Graphs, Factor And Remainder Theorem
12. Remainder Theorem
Let p(x) be a polynomial of degree greater than or equal to one, if p(x) is divided by (x - a). Then remainder R = p(a)
\(\therefore\)p(a) may be zero (or) non-zero real number.
Proof : If p(x) is divided by (x-a). we have obtained quotient q(x) and remainder r(x)
\(\therefore\) by division Algorithm
\(
p(x) = (x - a)q(x) + r
\)
if \(
x = a \Rightarrow p(a) = r
\)
For ex : Let \(
p(x) = x^2 - 3x + 5
\), if is divided by x - 2 x-2 = 0
\(
\therefore \,\,r = p(2) = 2^2 - 3.2 + 5 = 3
\) x = 2
\(
\therefore \,\,r = 3
\)
Remarks :
Divisor Remainder
x-a f(a)
x+a f(-a)
ax+b f(-b/a)
ax-b f(b/a)