Remainder Theorem

Remainder Theorem Info

When you divide a number you get the remainder. Ex: When you divide 11 by 2 you get 1 as the remainder (25 + 1 = 11). You wont get any remainder when the number is perfectly divided by the divisor. This is division of numbers, can we divide an equation? Answer is yes. Equations involving polynomials can be divided by a polynomial using the remainder theorem or the little Bézout’s theorem.

What is a Polynomial?

A polynomial is an algebraic expression made up of coefficients and the indeterminates. Other than division by variable, all other operations like addition, subtraction, multiplication, and positive integer exponents can be performed on polynomials. Division of a polynomial can be performed by dividing it by a linear degree polynomial. We make use of the remainder theorem for that. Let us learn the remainder theorem in detail now.

Remainder Theorem

Statement: According to the remainder theorem, if a polynomial, f(x), is divided by a linear polynomial, x – a, then the remainder will be the same as f(a). Alternatively, if you would like to evaluate the function f(x) for a given number, a, then you can divide that function by x – a and your remainder will be equal to the function f(a). 

Proof: f(x) is a polynomial of degree one or greater. Polynomial (x – a) is used to divide it, with ‘a’ as a real number. Let’s assume that the quotient is p(x) and the remainder is r(x). Thus, we can write, f(x) = (x – a)p(x) + r(x)

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Now, (x – a) is of 1degree. Additionally, r(x) is the remainder and as a result it has a degree less than its divisor: (x – a). Hence, r(x) has a degree of 0. This means that r(x) is a constant. The constant is called r. Hence,for any value of ‘x’, r(x) = r. Therefore,

f(x) = (x – a) p(x) + r

Now, let’s find f(a) or the value of f(x) at x = a.

f(a) = (a – a) p(a) + r

= (0)p(a) + r

= r

This shows that when a polynomial f(x) of a degree greater than or equal to one is divided by a linear polynomial (x – a), where a is a real number, the resulting remainder is r which is equal to f(a). In this way, the Remainder Theorem is proved.

Steps to Divide a Polynomial by a Non-Zero Polynomial 

  • Organize the given dividend and divisor of a polynomial in the descending order of its degree.
  • To get the first quotient term you need to divide the first term of the dividend by the first divisor term. 
  • Take the divisor and the first term of the quotient and multiply them. Subtract this product from the dividend to find the remainder. The remainder is the new dividend.
  • The above steps must be repeated until the new dividend’s degree is lower than the divisor’s degree.

Solved Examples of Remainder Theorem

Example 1: Find the reminder if x4 +2x2+4 is divided by (x – 2) and (x + 3)

Solution: Given (x-2) is a divisor of polynomial x4 +2x2+4 

(x – 2) = 0 x = 2.

By substituting x = 2 in the given polynomial we get

x4 +2x2+4  =  24 +(2)22+4 = 16 + 8 + 4 = 28

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So, Using the remainder theorem the remainder found is 28.

x4 +2x2+4 is divided by (x + 3)

(x + 3) = 0 x = -3.

By substituting x = -3 in the given polynomial, we get

x4 +2x2+4  =  (-3)4 + (-3)2+ 4 = 81 + 9 + 4 = 94

So, Using the remainder theorem the remainder found is 94.

For more problems on the remainder theorem or on any math topics you can subscribe to Cuemath classes. Visit their website for more details.

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