Here students will find some more questions on the Remainder and Factor Theorems. Students can find Important concepts on the ICSE Solutions For Class 10 Maths. For the need to understand the concepts in a better way, lets us see the most basic definitions.

### Explanation:

The **remainder theorem** tells us that for any polynomial f(x), if you divide it by the binomial x−a, the remainder is equal to the value of f(a). The **factor theorem** tells us that if a is a zero of a polynomial f(x), then (x−a) is a factor of f(x), and vice-versa.

For example, let’s consider the polynomial

f(x)=x2−2x+1

**Using the remainder theorem**

We can plug in 3 into f(x).

f(3)=32−2(3)+1

f(3)=9−6+1

f(3)=4

Therefore, by the remainder theorem, the remainder when you divide x2−2x+1by x−3 is 4.

You can also apply this in reverse. Divide x2−2x+1 by x−3, and the remainder you get is the value of f(3).

**Using the factor theorem**

The quadratic polynomial f(x)=x2−2x+1 equals 0 when x=1.

This tells us that (x−1) is a factor of x2−2x+1.

We can also apply the factor theorem in reverse:

We can factor x2−2x+1 into (x−1)2, therefore 1 is a zero of f(x).

Check out the ICSE Solutions Of Class 10 Maths Chapter 8 Remainder and Factor Theorems below: