Mathematics Class IX

Number Systems
Polynomials
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Remainder Theorem

If and are two polynomials in which the degree of degree of and are given then we can get the and so that:

where or degree of degree of .

It says that divided by , gives as quotient and as remainder.

Let’s understand it with an example

Division of a Polynomial with a Monomial

We can see that ‘x’ is common in the above polynomial, so we can write it as

Hence and the factors of

Steps of the Division of a Polynomial with a Non –Zero Polynomial

Divide by

Step 1:  Write the dividend and divisor in the descending order i.e. in the standard form. and

Divide the first term of the dividend with the first term of the divisor.

this will be the first term of the quotient.

Step 2: Now multiply the divisor with this term of the quotient and subtract it from the dividend.

Step 3: Now the remainder is our new dividend so we will repeat the process again by dividing the dividend with the divisor.

Step 4:

Step 5:

The remainder is zero.

Hence

Dividend = (Divisor Quotient) + Remainder

Remainder Theorem says that if is any polynomial of degree greater than or equal to one and let ‘t’ be any real number and is divided by the linear polynomial , then the remainder is .

As we know that

If is divided by then

If

To find the remainder or to check the multiple of the polynomial we can use the remainder theorem.

Example:

What is the remainder if a4 + a3 – 2a2 + a + 1 is divided by a – 1.

Solution:

P(x) = a4 + a3 – 2a2 + a + 1

To find the zero of the (a – 1) we need to equate it to zero.

a -1 = 0

a = 1

p (1) = (1)4 + (1)3 – 2(1)2 + (1) + 1

= 1 + 1 – 2 + 1 + 1

= 2

So by using the remainder theorem, we can easily find the remainder after the division of polynomial.

Factor Theorem

Factor theorem says that if p(y) is a polynomial with degree n 1 and t is a real number, then

1. (y – t) is a factor of p(y), if p(t) = 0, and
2. P (t) = 0 if (y – t) is a factor of p(y).

Example: 1

Check whether is the factor of  using factor theorem.

Solution:

According to the factor theorem if is the factor of then , as the root of is 3.

Hence, is the factor of .

Example: 2

Find the value of k, if is a factor of

Solution:

As is the factor so

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