Question

Using remainder theoram,find the remainder when x⁴-3x³+2x²-4
is divided by x+2
​

Answer 1

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Answer 2

Given :-

x⁴ - 3x³ + 2x² - 4

( x - 2 ) when divides leaves remainder

Required to find :-

• Find the remainder ?

Method used :-

>> Remainder Theorem <<

Solution :-

Given information :-

x⁴ - 3x³ + 2x² - 4

( x + 2 ) when divides leaves remainder

we need to find the remainder theorem

So,

Let's consider the given polynomial as ;

p ( x ) = x⁴ - 3x³ + 2x² - 4

( x + 2 ) when divides p ( x ) it leaves remainder

So,

Let,

x + 2 = 0

x = - 2

p ( - 2 ) = ( - 2 )⁴ - 3 ( - 2 )³ + 2 ( - 2 )² - 4

p ( - 2 ) = 16 - 3 ( - 8 ) + 2 ( 4 ) - 4

p ( - 2 ) = 16 + 24 + 8 - 4

p ( - 2 ) = 48 - 4

p ( - 2 ) = 44

Therefore ,

When ( x + 2 ) divides p ( x ) it leaves remainder as 44

Additional Information :-

The remainder theorem states that when f ( x ) is divided by ( x - a ) the remainder will be equal to the value which we get by substituting this value of " a " in place of x in p ( x )

This is represented as ;

p ( x ) ÷ ( x - a ) = x

p ( a ) = x

Here, the value is same in both cases

Example ,

p ( x ) = x² - 2x + 1 ÷ ( x - 1 )

So,

When ( x + 1 ) divides p ( x ) the remainder is zero

Now substitute the 1 in value of p ( x )

So,

p ( 1 ) = ( 1 )² - 2(1) + 1

p ( 1 ) = 1 - 2 + 1

p ( 1 ) = 2 - 2

p ( 1 ) = 0

Hence,

In this both cases the value is same .

This is defined in the remainder theorem .