**Remainder Theorem (शेषफल प्रमेय)**

**Remainder Theorem (शेषफल प्रमेय)**

*Let p(x) be any polynomial of degree greater than or equal to 1 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. Suppose that when p(x) is divided by x – a, then *quotient* is q(x) and remainder is r(x). i.e.*

*Since the degree of (x – a) is 1 then the degree of r(x)is less than the degree of x – *a,* the degree of r(x) = 0. This means that r(x) is a constant, say r.*

*So for every value of x, r(x) = r.*

*therefore, p(x) = (x – a)q(x) + r*

*If x = a, then the equation will give us:*

*p(a) = (a – a)q(a) + r = 0 + r*

*p(a) = 0*

*Which proves the theorem.*

### Polynomial Remainder Theorem Examples With Answers

**Example 1:-**

**Find the remainder when is divided by x – 1?**

** को x – 1 से भाग देने पर शेषफल पता कीजिए?**

**Explanation:**

*Here p(x) = and the zero of x – 1 is 1.*

*Therefore by remainder theorem we can say that the remainder will be p(1).*

*p(1) =*

*p(1) = *

*So the remainder will be 4.*

**Example 2:-**

**Find the remainder when is divided by x + 1?**

** को x + 1 से भाग देने पर शेषफल पता कीजिए?**

**Explanation:**

*Here p(x) = and the zero of x + 1 is -1.*

*Therefore by remainder theorem we can say that the remainder will be p(-1).*

*p(-1) = *

*p(-1) = *

*So the remainder will be 9.*

**Example 3:-**

**Check whether x – 2 is a factor of **

**जांच कीजिए कि **x – 2,** का एक गुणानखण्ड है या नहीं?**

**Explanation:**

*If x – 2 is a factor of then when we will divide by x – 2 then remainder must be zero.*

*So by remainder theorem:*

*p(x) = and the zero of x – 2 is 2.*

*So by remainder theorem we can say that the remainder will be p(2).*

*p(2) = *

*= *

= 0

*So there remainder is zero that means x – 2 is a factor of *

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