An important question from examination point of as it has been already asked in previous year paper of 2005 in which we are asked to find the values of a and b if (x – 2) is a factor of the expression x3+ax2+bx+6. when this expression is divided by (x – 3) , it leaves the remainder 3
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(x – 2) is a factor of the expression x 3 +ax 2 +bx+6. when this expression is divided by (x – 3) , it leaves the remainder 3. find the values of a and b.
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Let p(x) = x3 + ax2 + bx +6
(x-2) is a factor of the polynomial x3 + ax2 + b x +6
p(2) = 0
p(2) = 23 + a.22 + b.2 +6 =8+4a+2b+6 =14+ 4a+ 2b = 0
7 +2 a +b = 0
b = – 7 -2a -(i)
x3 + ax2 + bx +6 when divided by (x-3) leaves remainder 3.
p(3)=3
p(3) = 33 + a.32 + b.3 +6= 27+9a +3b +6 =33+9a+3b = 3
11+3a +b =1
=> 3a+b =-10
=> b= -10-3a -(ii)
Equating the value of b from (ii) and (i) , we have
(- 7 -2a) = (-10 – 3a)
a = -3
Substituting a = -3 in (i), we get
b = – 7 -2(-3)
= -7 + 6
= -1
Thus the values of a and b are -3 and -1 respectively.