Question

If ax3+3x2+bx-3 has a factor 2x+3 and leave -3 when divided by x+2 find the value of a and b

Answer 1

p(x) = ax³ + 3x² + bx - 3

g(x) = 2x + 3 or x = -3/2

f(x) = x + 2 or x = -2

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

=> -8a + 12 - 2b = 0

=> -8a - 2b = -12

=> -(8a + 2b) = -12

=> 8a + 2b = 12

=> 4a + b = 6 ------------- (i)

p(-3/2) = a(-3/2)³ + 3(-3/2)² + b(-3/2) - 3 = 0

=> -27/8 a + 27/4 -3/2 b = -3

=> (-27a + 54 - 6b)/8 = -3

=> -27a + 54 - 6b = -24

=> -27a - 6b = -24 - 54

=> -27a - 6b = - 78

=> -3(9a + 2b) = -78

=> 9a + 2b = 26 -----------(ii)

Multiplying eq.(i) by 2 and subtracting it from eq(ii) , we have

8a + 2b = 12

9a + 2b = 26

----------------------

-a = - 14

[ a = 14 ]

Putting value of a in eq(i)

4a + b = 6

4(14) + b = 6

b = 6 - 56

b = - 51