Polynomial is an algebraic expression consisting of variable and coefficient. Variable is also at times called indeterminate. We can perform any of the operations using polynomials whether it be multiplication, division, subtraction or addition. Examples of polynomial with one variable are x^{2 }+ x – 8, y^{3 }+ y^{2 }– 52, z^{2}+64

The word polynomial was derived from the Greek word ‘**poly**’ meaning ‘many’ and ‘**nominal**’ meaning ‘terms’, so altogether it is said as “**many terms**”. A polynomial can not have infinite terms.

**Remainder Theorem**

Let **g(x)** be a polynomial of degree 1 or greater than 1 and let b be any real number. If **g(x)** is divided by the linear polynomial x – b, then the remainder is **p(b)**.

**Proof**

Let g(x) be a polynomial with degree 1 or greater than 1. Suppose that when g(x) is divided by (x – b), the quotient is q(x) and the remainder is r(x), i.e.,

g(x) = (x – b) q(x) + r(x) ……(1)

Since the degree of x – b is 1 and that of r(x) is less than the degree of x – b, the degree of r(x) = 0. This means that r(x) is constant. Thus, we can write the equation (1) as

g(x) = (x – b) q(x) + r ……(2)

In particular, if x = b, then equation (2) becomes

p(b) = (b – b) q(b) + r => r

### Sample Problems on Remainder Theorem

**Problem 1: Find the remainder when g(x) = x ^{4} – x^{3} + x^{2} – 2x + 1 is divided by x – 2.**

**Solution:**

Zero of x – 2 is 2, so as per the remainder theorem

Remainder, in this case, will be g(2).

So,

g(2) = (2)

^{4}– (2)^{3}+ (2)^{2}+ 2(2) + 1 = 17

**Problem 2: Find the root of the polynomial x ^{2} – 5x + 4**

**Solution: **

The approach of solving such questions include to choose a number in such a manner, that putting it to use, will yield a zero remainder.

f(x) = x

^{2 }– 5x + 4f(4) = 4

^{2 }– 5(4) + 4f(4) = 20 – 20 = 0

So, (x – 4) must be a factor of x

^{2 }– 5x + 4

**Problem 3: Find the remainder when t ^{3} – 2t^{2 }+ 4t + 5 is divided by t – 1.**

**Solution: **

Since, here it is already given that we need to find the remainder when the given quotient is divided by t – 1. So, accordingly we will put 1 in place of x, to solve and get the remainder.

Here, p(t) = t

^{3}– 2t^{2 }+ 4t + 5, and the zero of t – 1 is 1∴ g(1) = (1)

^{3}– 2(1)^{2}+ 4 + 5 = 8By the Remainder Theorem, 8 is the remainder when t

^{3 }– 2t^{2 }+ 4t + 5 is divided by t – 1

**Problem 4: Find the remainder when x ^{3 }–^{ }x^{2 }+ 2 is divided by x – 2.**

**Solution:**

Here, we will put x = 2 in the given quotient to find the remainder

2^3 – 2^2 + 2

= 8 – 4 + 2

= 6

By the Remainder Theorem, 6 is the remainder when x

^{3}– x^{2 }+ 2 is divided by x – 2.

**Problem 5: By what should, x ^{3} – x^{2} – 4 be divided to give 0 as remainder.**

**Solution:**

We will use Hit & Trial method to find the answer,

Clearly putting x = 2, will give zero as remainder.

So, this will be the answer to yield a zero remainder on division by x – 2.

## Recommended Posts:

- Pythagoras Theorem and its Converse - Triangles | Class 10 Maths
- Theorem - The lengths of tangents drawn from an external point to a circle are equal - Circles | Class 10 Maths
- General and Middle Terms - Binomial Theorem - Class 11 Maths
- Theorem - The tangent at any point of a circle is perpendicular to the radius through the point of contact - Circles | Class 10 Maths
- Mid Point Theorem - Quadrilaterals | Class 9 Maths
- Mean value theorem - Advanced Differentiation | Class 12 Maths
- Class 9 NCERT Solutions - Chapter 2 Polynomials - Exercise 2.3
- Class 9 NCERT Solutions - Chapter 2 Polynomials - Exercise 2.1
- Class 10 NCERT Solutions - Chapter 2 Polynomials - Exercise 2.2
- Multiply two polynomials
- Program to add two polynomials
- Adding two polynomials using Circular Linked List
- Multiplication of two polynomials using Linked list
- Adding two polynomials using Linked List
- Arithmetic Progression - Common difference and Nth term | Class 10 Maths
- Mensuration - Area of General Quadrilateral | Class 8 Maths
- Mensuration - Volume of Cube, Cuboid, and Cylinder | Class 8 Maths
- Area of a Triangle - Coordinate Geometry | Class 10 Maths
- Distance formula - Coordinate Geometry | Class 10 Maths
- Algebraic Expressions and Identities | Class 8 Maths

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.