Question

plz help me out with the 25th one

Answer 1

Answer:

a = -3, b = -1

Step-by-step explanation:

Since x - 2 is a factor of

( x ^3 + ax ^2 + bx + 6 ),

x = 2 in the function of the polynomial will give 0 as the answer.

Thus,

( 2 ) ^3 + ( a ) ( 2 ) ^2 + ( b ) ( 2 ) + 6 = 0

8 + 4a + 2b + 6 = 0

4a + 2b + 14 = 0

2a + b + 7 = 0

2a + b = -7 ------- ( 1 )

Since x - 3 leaves a remainder 3,

x = 3 in the function of the polynomial will give 3 as the answer.

Thus,

( 3 ) ^3 + ( a ) ( 3 ) ^2 + ( b ) ( 3 ) + 6 = 3

27 + 9a + 3b + 6 = 3

9a + 3b + 33 = 3

3a + b + 11 = 1

3a + b = -10 ---- ( 2 )

By ( 2 ) - ( 1 ),

( 3a + b ) - ( 2a + b ) = -10 - ( -7 )

a = -10 + 7

a = -3

Thus, in ( 1 ),

2 ( -3 ) + b = -7

-6 + b = -7

b = -7 + 6

b = -1