Assume that x-2 is a factor of the polynomial f(x)=x^3+ax^2+bx+2 and that f(x)gives a remainder of -3 when it is divided by x+1. Then a =?,b=?

Question

431 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-18T14:37:12

The polynomial is #=x^3-3x^2+x+2#

By the remainder theorem

#f(x)=(x-c)q(x)+r#

#f(x)=x^3+ax^2+bx+2#

#(x-2)# is a factor of #f(x)#

#f(2)=8+4a+2b+2=0#

#2a+b=-5#............................#(1)#

Division by #(x+1)#

#f(-1)=-1+a-b+2=-3#

#a-b=-4#............................#(2)#

Solving the simultaneous equatins #(1)# and #(2)#

#{(2a+b=-5),(a-b=-4):}#

#<=>#, #{(2a+b=-5),(3a=-9):}#

#<=>#, #{(b=1),(a=-3):}#

Finally,

#f(x)=x^3-3x^2+x+2#