Question

Find the remainder when polynomial 3x^3-4x^2+ 7x -5 is devided by (x-3) (x+3)

Answer 1

p(x)=3x³-4x²+7x-5

let, x-3=0

x=3.

p(3)=3(3)³-4(3)²+7(3)-5

=3(27)-4(9)+21-5

=81-36+21-5

=102-41

=61

letx+3=0

x=-3

p(-3)=3(-3)³-4(-3)²+7(-3)-5

=3(-27)-4(9)-21-5

=-81-36-21-5

=-143

Answer 2

Given :-

Qubic polynomial :- 3x³ - 4x² + 7x - 5

Required to find :-

Remainder when the given cubic polynomial is divided by ( x - 3 ) & ( x + 3 )

Method used :-

• Remainder theorem

Solution :-

Given information :-

Qubic polynomial : 3x³ - 4x² + 7x - 5

we need to find the remainder when the given cubic polynomial is divided by ( x - 3 ) & ( x + 3 )

Let's consider the given cubic polynomial as ;

p ( x ) = 3x³ - 4x² + 7x - 5

( x - 3 ) when divides p ( x ) it leaves remainder

So,

Let;

=> x - 3 = 0

=> x = 3

Substitute the value of x in p ( x )

p ( 3 ) = 3 ( 3 )³ - 4 ( 3 )² + 7 ( 3 ) - 5

p ( 3 ) = 3 ( 27 ) - 4 ( 9 ) + 21 - 5

p ( 3 ) = 81 - 36 + 21 - 5

p ( 3 ) = 102 - 41

p ( 3 ) = 61

Hence,

when ( x - 3 ) divides p ( x ) the remainder is 61 .

Similarly ,

( x + 3 ) when divides p ( x ) it leaves some remainder

So,

Let;

=> x + 3 = 0

=> x = - 3

p ( x ) = 3x³ - 4x² + 7x - 5

Substitute this value in place of x in p ( x )

p ( - 3 ) = 3 ( - 3 )³ - 4 ( - 3 )² + 7 ( - 3 ) - 5

p ( - 3 ) = 3 ( - 27 ) - 4 ( 9 ) - 21 - 5

p ( - 3 ) = - 81 - 36 - 21 - 5

p ( - 3 ) = - 143

Hence,

when ( x + 3 ) divided p ( x ) the remainder is

- 143

Therefore ;

when ( x - 3 ) & ( x + 3 ) divided p ( x ) the remainders are 61 & - 143

Extra dose :-

• What is remainder theorem ?

☛ The remainder theorem states that ;

☛ when ( x - a ) divides a polynomial p ( x ) the remainder is equivalent to p ( a ) .

☛ This is only applicable when ( x - a ) is a linear polynomial .

☛ However, when the remainder comes to be as zero then we can say that the given linear polynomial is the factor of that p ( x )

☛ Actually, the factor theorem is derived from this remainder theorem