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Remainder Theorem

You might have learned division in your school. Let us divide 36 by 7, we get the quotient 5 and reminder 1. It can be written as 36 = (75) + 1. Here reminder 1 is less than the quotient 5. Can we divide the polynomials from a polynomial? Yes. Using the remainder theorem it is easy to divide the polynomials. It is also known as little Bézout’s theorem.

What is a Remainder Theorem?

You already know that the remainder theorem is used for polynomial division. So, the remainder theorem can be stated as, If a polynomial p(x) of degree greater than one or equal to one, is divided by a linear polynomial (x-t). Then the reminder formed is equal to p(t). where t represents any real number.

What is a Polynomial?

A polynomial is an algebraic expression that is formed by the indeterminates and coefficients. Operations like addition, subtraction, multiplication, and exponentiation of variables of whole numbers are used to form the polynomials. Ex: x2 – 4. The remainder theorem is used to find the remainder formed by dividing these polynomials by linear polynomials. You can learn more about polynomials on the Cuemath website.

Proof for Polynomial Remainder Theorem:

You know that p(x) = (x – t). q(x) + r(x) Where r(x) is a reminder, p(x) is a dividend, (x – t) is a divisor and q(x) is a quotient. Here the degree of r(x) is less than the degree of (x – t) and (x – t) is a polynomial of degree 1. A Polynomial of degrees less than 1 is a constant. Hence r(x) = constant or 0.

Let r(x) = constant r

∴ p(x) = (x – t). q(x) + r

Now replace x by t

∴ P(t) = (t – t). q(t) + r, P(t) = 0 + r

p(t) = r

Hence proved.

Divide a Polynomial by a Non-Zero Polynomial Using the Following Steps

1. Arrange the dividend and divisor of a polynomial in the decreasing order of its degree.

2. Divide the first term of the dividend by the first divisor term to get the first quotient term.

3. Take the product of the divisor and the first term of the quotient. The remainder is formed by subtracting this product from the dividend. The reminder so formed is the new dividend now.

4. Repeat the above-mentioned steps until you get the degree of the new dividend less than the degree of the divisor.

Solved Examples of Remainder Theorem:

1. Find the reminder if x3 +2x2+4x+5 is divided by (x-3)

Solution: Given – (x – 3) is a divisor of polynomial x3 +2x2+4x+5

(x – 3) = 0  x = 3.

Substitute the value of x in the given polynomial.

 x3 +2x2+4x+5 = 23 + 2 22 + 4 2 + 5 = 8 + 8 + 8 + 5 = 29.

So, Using the remainder theorem the remainder found is 29.

2. Find the value of t if (x – t) is the divisor of x2 – 2tx + 4.

Solution: If (x – a) is a divisor of x2 – 2tx + 4 then P(a) = 0

P(a) = t2 – 2  t t + 4 = 0  t2 – 2t2 +4 = 0

-t2 = – 4  t = 2.

Hence the value of t is 2.

Let me end it here. Hope you understood the concept well. If you still have queries, please visit cuemath website for a more detailed explanation. You can also learn more math concepts in Cuemath in an interesting way. If you have understood the theory of remainders then let me ask you a question. What is the Euler remainder theorem? Try to find an answer to this question, you can check in Cuemath website also.