Remainder Theoram
Last updated at August 13, 2026 by Teachoo
Let us first divide 7 by 3,
7/3
Here
7 = 3 × 2 + 1
i.e. Dividend = Divisor × Quotient + Remainder
We can do the same when dividing two polynomials
In Example 6 ,
We divided 3x 2 + x − 1 by 1 + x
Our division was
Here
Dividend = 3x 2 + x – 1
Divisor = x + 1
Quotient = 3x – 2
And Remainder = 1
Dividend = Divisor × Quotient + Remainder
3x 2 + x – 1 = (x + 1) (3x – 2) + 1
3x 2 + x – 1 = x (3x – 2) + 1 (3x – 2) + 1
3x 2 + x – 1 = 3x 2 – 2x + 3x – 2 + 1
3x 2 + x – 1 = 3x 2 + x – 1
Transcript
Dividing two numbersQuotient Divisor Dividend Remainder Which can be rewritten as a sum like this: Division Algorithm is Dividend = Divisor × Quotient + Remainder Quotient Divisor Dividend Remainder Dividing two Polynomials Let’s divide 3x2 + x − 1 by 1 + x We can write Dividend = Divisor × Quotient + Remainder 3x2 + x – 1 = (x + 1) (3x – 2) + 1 What if…We don’t divide? We divide 3x2 + x − 1 by x + 1 So, Dividend = Divisor × Quotient + Remainder 3x2 + x – 1 = (x + 1) × Quotient + Remainder Putting x = −1 3(−1)2 + (−1) – 1 = (−1 + 1) × Quotient + Remainder 1 = Remainder Remainder = 1
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