Question

Verify if -1/2 and 5/2 are zeroes of the
polynomial 4x3 - 21x - 10. If yes, then
factorise the polynomial (CBSE 2015)​

Answer 1

Answer:

Given polynomial in factorised form is ( 2x + 1 )( 2x - 5 )( x + 2 )

Step-by-step explanation:

If the given fractions { - 1 / 2 and 5 / 2 } are the zeroes of 4x^3 - 21x - 10, then the result must be 0 on substituting - 1 / 2 and 5 / 2 in place of x.

Checking by substituting the values :

If - 1 / 2 is a factor of 4x^3 - 21x - 10 :

= > 4x^3 - 21x - 10

= > 4( - 1 / 2 )^3 - 21( - 1 / 2 ) - 10

= > 4( - 1 / 8 ) - 21( - 1 / 2 ) - 10

= > - 1 / 2 + 21 / 2 - 10

= > ( - 1 + 21 - 20 ) / 10

= > ( - 21 + 21 ) / 10

= > 0 / 10

= > 0

As the result is 0, - 1 / 2 is a zero of the given polynomial.

If 5 / 2 is a factor of the given polynomial :

= > 4x^3 - 21x - 10

= > 4( 5 / 2 )^3 - 21( 5 / 2 ) - 10

= > 4( 125 / 8 ) - 21( 5 / 2 ) - 10

= > 125 / 2 - 105 / 2 - 10

= > ( 125 - 105 - 20 ) / 10

= > ( 125 - 125 ) / 10

= > 0 / 10

= > 0

As the result is 0, 5 / 2 is a zero of the given polynomial.

Factorisation of the given polynomial :

= > 4x^3 - 21x - 10

= > 4x^3 + 2x^2 - 2x^2 - 21x - 10

= > 2x^2( 2x + 1 ) - 2x^2 - ( 1 + 20 )x - 10

= > 2x^2 ( 2x + 1 ) - 2x^2 - x - 20x - 10

= > 2x^2( 2x + 1 ) - x( 2x + 1 ) - 10( 2x + 1 )

= > ( 2x + 1 )( 2x^2 - x - 10 )

= > ( 2x + 1 ){ 2x^2 - ( 5 - 4 )x - 10 }

= > ( 2x + 1 ){ 2x^2 - 5x + 4x - 10 }

= > ( 2x + 1 ){ x( 2x - 5 ) + 2( 2x - 5 ) }

= > ( 2x + 1 )( 2x - 5 )( x + 2 )

Hence the given polynomial in factorised form is ( 2x + 1 )( 2x - 5 )( x + 2 )