The expression 2x^3-x^2+ax+b gives a remainder of 14 when divided by(x-2) and a remainder of -86 when divided by (x+3)/Find the values of a and b?

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Answer 1 · 2018-06-14T18:23:41

#a = 5, b = -8#

#P(x) = 2x^3 - x^2 + ax + b#

#P(2) = 14# and #P(-3) = -86#

#((2*8 - 4 + 2a + b = 14), (-2*27 - 9 -3a + b = -86))#

#((2a + b = 2), (-3a + b = -23))#

#2a + 3a = 2 + 23#

#5a = 25#

#a = 5#

#2a + b = 2#

#10 + b = 2#

#b = -8#

Answer 2 · 2018-06-14T18:30:05

# a=5, and, b=-8#.

Prerequisite : The Remainder Theorem (RT) :

If a polynomial #p(x)# is divided by #(x-alpha)#, then the

remainder is #p(alpha)#.

Let, #p(x)=2x^3-x^2+ax+b#.

Given that, #p(x)#, when divided by #(x-2)#, leaves the remainder #14#

Hence, by RT, #p(2)=14 rArr 2(2)^3-2^2+a(2)+b=14,#

# or, 2a+b=2.....................................................(1)#.

Similarly,

#p(-3)=-86 rArr -3a+b=-23.....................(2)#.

Solving #(1) and (2), a=5, and, b=-8#.