Question

If x² - 3 x + 2 is a factor of x⁴ - px² + q, then the values of p
and q are
(a) 5, -4 (b) 5, 4 (c) -5, 4 (d)-5, -4.

Answer 1

Answer:

5 , 4

b). option is correct.

Step-by-step explanation:

Given :

p ( x ) =  x⁴ - p x² + q

g ( x ) = x² - 3 x + 2

Also g ( x ) is factor of p ( x )

x² - 3 x + 2

= > x² - 2 x - x + 2

= > x ( x - 2 ) - ( x - 2 )

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

Zeroes of g ( x ) :

= > x = 2  OR x = 1

Since g ( x ) is factor of p ( x ) putting values there :

p ( x ) =  x⁴ - p x² + q

Putting x = 1

p ( 1 ) = 1⁴ - p . 1² + q

p (  1 ) = 1 - p + q

Equating with zero :

1 - p + q = 0

p = 1 + q  

Multiply by 4 both side :

4 p = 4 + 4 q .... ( i )

p ( x ) =  x⁴ - p x² + q

Putting x = 2

p ( 2 ) = 2⁴ - p . 2² + q

p ( 2 ) = 16 - 4 p + q

Equating with zero :

16 - 4 p + q = 0

4 p = 16 + q ... ( ii )

From ( i )  and ( ii ) we get :

4 + 4 q = 16 + q

= > 4 q - q = 16 - 4

= > 3 q = 12

= > q = 4

Putting q = 4 in ( i ) we get :

4 p = 4 + 4 q .... ( i )

= > 4 p = 4 + 16

= > 4 p = 20

= > p = 5

Therefore , we get required answer.