Remainder theorem explained with examples

Learn and know what is the meaning of remainder theorem in mathematics and how and why to use this theorem. We will study about remainder theorem in the class 9 in algebra chapter.

Remainder theorem explained with examples

In class 9, we have two main theorems i.e. remainder theorem and factor theorem. These two theorems are important and very useful. Now we will discuss about the remainder theorem and if we observe the name, it is understandable that we need to find the remainder. But the problem is how to find the remainder. So to know this we have to study and learn remainder theorem.

What is remainder theorem in maths?

Let us consider a polynomial f(x) with the degree greater than 1 or equal to 1 and let “a” be a real number then if the polynomial f (x) is divided by the (x-a) then the remainder is given as f(a).

Note:

If suppose f (x) is divided by (x+a) then the remainder is equal to the value of f (-a)

Use of learning the remainder theorem:

In some cases we may be dividing a polynomial (with degree more than 1 or equal to 1) with a linear polynomial and after dividing at last we will get the remainder. It is clear that in this process finding the remainder is somewhat difficult and also taking much time. So avoid this if you learn what is remainder theorem? Then you can apply this theorem and easily we can find the remainder.

Examples:

Find the reminder when f(x) = 2 { x }^{ 3 } { x }^{ 2 } + 6x + 9 is divided by the polynomial x-2.

Solution:

According to the remainder theorem, the remainder of the polynomial f(x) is f(2).

So, remainder = f (2) = 2 × { 2 }^{ 3 } { 2 }^{ 3 } + 6 × 2 + 9 = 16 – 4 +12 + 9 = 37 – 4 = 33.

What is the remainder if g(y) = { y }^{ 4 } + 6 { y }^{ 2 } – 2y + 10 is divided by y + 2.

Solution:

According to the remainder theorem, the remainder of the polynomial g(y) is g(-2).

So, remainder = g (-2) = { (-2) }^{ 4 } + 6 × { (-2) }^{ 2 } – 2 × (-2) + 10 = 16 + 24 + 4 + 10 = 54.

FAQ’s on the remainder theorem:

If g (x) is a polynomial and it is divided by cx+d then what is remainder?

Given that g (x) is a polynomial and cx+d is a linear polynomial which is dividing g (x). Then the remainder is given as g ( \frac { -d }{ c } ).

In the above example, how we got remainder as g ( \frac { -d }{ c } )?

We have to make divisor equal to zero and find x value. So in the example, divisor is cx+d.

So, cx+d = 0

     cx = -d

    x = \frac { -d }{ c }

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