Question

if x-2 is a factor of x^3+px^2+qx+16 and p-q=18,find the value of p and q?

Answer 1

Step-by-step explanation:

x-2 is a factor, x = 2

p-q = 18

p=18+q

x³ + px² + qx + 16 =0

2³ + p(2)² + q(2) + 16 = 0

8 + (18 + q)4 + 2q + 16 = 0

8 + 72 + 4q + 2q + 16 =0

80 + 6q + 16 = 0

6q + 96 = 0

6q = -96

q = -96/6

q=-16

p = 18 + q

= 18+(-16) = 18-16 ==> p = 2

Answer 2

Answer:

Step-by-step explanation:

Let f ( x ) = x³ + px² + qx + 16 = 0

If ( x - 2 ) is a factor of f( x ),then x = 2.

On substituting ,

f ( 2 ) = ( 2 )³ + p ( 2 )² + q ( 2 ) + 16 = 0

⇒ 8 + 4p + 2q + 16 = 0

⇒ 24 + 4p + 2q = 0

⇒ 12 + 2p + q = 0 ( ∵ taken by 2 as common )

⇒ 2p + q = - 12-------eq ( 1 ) (say )

Let p - q = 18--------eq ( 2 )

Now adding eq( 1 ) + eq ( 2 ) ,we get

2p + q + p - q = - 12 + 18

3p = 6

p = 6/3 = 2.

Now, substituting p = 2 in the eq ( 1 ) , we have

2( 2 ) + q = - 12

4 + q = - 12

p = - 12 - 4 = - 16

∴ The values of p and q are 2 and - 16.