Question

If x-6 is a factor of x^3+ax^2+bx'b=0and a-b=7.find a and b

Answer 1

Let p(x) = x³ + ax² + bx - b

As (x- 6) is a factor of p(x) then after substituting the value x= 6 we will get 0 i.e. p(6) = 0

Now,

p(6) = (6)³ + a(6)² + b(6) - b

= 216 + 36a + 6b -b

= 216 + 36a + 5b

= 36a + 5b = -216 ---------(1)

Also, we have

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

a = 7 + b

put a = 7 + b in eq (1)

=> 36 (7 + b) + 5b = -216

=> 252 + 36b + 5b = -216

=> 252 + 41b = -216

=> 41b = -216 -252

=> 41b = - 468

=> b = - 468 / 41

Now, put the value of b in eq (2)

=> a - (-468 / 41) = 7

=> (41a + 468) / 41 = 7

=> 41a + 468 = 7 × 41

=> 41a + 468 = 287

=> 41a = 287 - 468

=> 41a = - 181

=> a = - 181 / 41

Hence, a = -181/41 and b = -468/41

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