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Remainder Theorem

Types of Polynomials Assignment / Homework Help
Dividend = (Divisor Quotient) + Remainder

If p(x) and t(x) are two polynomials such that degree of p(x) ≥ degree of t(x) and t(x) ≠ 0, then polynomials q(x) and r(x) can be found such that:

p(x) = t(x) q(x) + r(x),

Where :

r(x) = 0 or degree of r(x) < degree of t(x)

Or

It can be said that p(x) divided by t(x) gives q(x) as quotient and r(x) as remainder.


Statement of remainder theorem:

"Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x a, then the remainder is p(a)."

Proof:

Let p(x) be any polynomial with degree of p(x) ≥ 1. Assuming that when p(x) is divided by x- a, the quotient is q(x) and the remainder is r(x), i.e. ,

p(x) = (x a) q(x) + r(x)

Since the degree of x a is 1 and the degree of r(x) is less than the degree of q(x), degree of r(x) = 0. This implies that r(a) is a constant, say r.

Hence for every value of x, r(x) = r.

Hence p(x) = (x a)q(x) + r

If x = a, then the above equation will be:

p(a) = (a a)q(a) + r

       = r (hence proved)

Example:

Find the remainder when x3 + 2x2 + x + 1 is divided by x + 1.

Solution:

Let p(x) = x3 + 2x2 + x + 1

x + 1 = 0 x = -1


Hence remainder = p(-1) = (-1)3 + 2(-1)2 + (-1) + 1
= -1 + 2 -1 + 1
= 1
Example:

Using long division method, divide the polynomial 3x4 4x3 3x - 1 by x 1.


Solution:

3x3 - x2 x - 4
x 1 3x4 4x3 3x 1
3x4 3x3
-    +
- x3 3x 1
- x3 + x2
+    -
-x2 - 3x -1
-x2 + x
+    -
- 4x 1
- 4x + 4
+    -
- 5

Hence Quotient = 3x3 - x2 x 4 and Remainder = -5.


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