Question #11033

polynomial f(x)= x4-2x3+3x2-ax+b when divided by (x-1) and (x+1) leaves reminder 5 and 19.Find the values of A and B and the reminder when f(x) is divided by (x-2)

Expert's answer

we use theorem f(x) divided by x-c leaves reminder which equal to f(a)

so

f(1)=5, f(-1)=19

1^4-2*1^3+3*1^2-a*1+b=5

and

(-1)^4-2*(-1)^3+3*(-1)^2-a*(-1)+b=19

2-a+b=5 and 6+a+b=19 so a=b-3

from first equation

6+b-3+b=19

b=8,

a=5,

f(x)=x^4-2x^3+3x^2-5x+8

and reminder when f(x) is divided by x-2 is

f(2)=10

so

f(1)=5, f(-1)=19

1^4-2*1^3+3*1^2-a*1+b=5

and

(-1)^4-2*(-1)^3+3*(-1)^2-a*(-1)+b=19

2-a+b=5 and 6+a+b=19 so a=b-3

from first equation

6+b-3+b=19

b=8,

a=5,

f(x)=x^4-2x^3+3x^2-5x+8

and reminder when f(x) is divided by x-2 is

f(2)=10

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