the expression 2x^3 - x^2 +ax+ b gives a remainder 14 when divided by (x-2) and a remainder -86 when divided by (x+3).find the values of a and b..

Question

it is yet another a remainder theorem problem i request u to solve it....thankyou very much

Answer 1

Hi Rodela,

 

2x³-x²+ax+b  gives a remainder 14 when divided by (x-2)

 

so substitute x=2 in the given expression

 

2(2)³-(2)²+2a+b=14

16-4+2a+b=14

12+2a+b=14     

2a+b=2     equation 1

 

2x³-x²+ax+b gives a remainder -86 when divided by  (x+3)

 

so substitute x=-3 in the given expression

 

2(-3)³-(-3)²+(-3)a+b=-86

-54-9-3a+b=-86

-63-3a+b=-86

-3a+b=-23    equation 2

 

subtracting  equation 2 from equation 1

 

5a= 25

a=5

 

substitute the value of a=5 in equation 1

2(5)+b=2

b=-8

 

The value of a=5 and b=-8

 

Answer 2

This problem can be approached using either synthetic division or long division of a polynomial by a binomial.

 

Dividing 2 x³ - x² +ax + b  by  (x-2)  gives a remainder of 12 + 2 a + b.

Dividing 2x³ - x² + a x + b  by (x+3)   gives a remainder of -63 - 3a + b

 

Since the first remainder must be 14 and the second be -86   , we have two equations and two unknowns:

 

12 + 2 a + b  = 14   and

-63 - 3a +b   =  -86

 

The solution to this set of equations is  a = 5  and b = -8