# Remainder Theorem

Let us consider two polynomials and

Where

Â Â Â Â Â Â Â Â Â Â =

Divide by = by long division method

**Step 1:** Arrange the dividend and the divisor in the standard form.

**Step 2:** Divide the first term of the dividend by the first tern of the divisor.

first term of the quotient

**Step 3:**

Then we multiply the quotient and the divisor and subtract it from the dividend. Then we treat the remainder as the dividend and repeat the step 2.

**Step 4:**

We continue the process till the degree of the remainder is less than the degree of the divisor.

**Step 5: **

Add all the quotients we get, .

In general, and are two polynomials such that degree of and , then we can find polynomials and such that where, = 0 or degree of

Remainder Theorem:

Let be any polynomial with degree greater than or equal to one and let ** a ** be any real number. If is divided by the linear polynomial then the remainder is.

**Proof:**

Let be a polynomial with degree greater than or equal to 1.

Suppose is divided by with quotient and remainder then,

= +

Since degree of is 1 and degree of is less than 1, is a constant say, r.

Therefore, = +

Â Â Â If

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â =

Â

**Example :** Find the remainder when divided by

Here =

Zero of is 4

Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â = **= **212

The remainder when divided by is 212.

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