SOLUTION: Given that f(x)=x^3+ax^2+bx-3 leaves remainders 1 and -9 when divided by x-1 and x+1. Calculate the values of a and b

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Given that f(x)=x^3+ax^2+bx-3 leaves remainders 1 and -9 when divided by x-1 and x+1. Calculate the values of a and b       Log On


   

Question 1187763: Given that f(x)=x^3+ax^2+bx-3 leaves remainders 1 and -9 when divided by x-1 and x+1. Calculate the values of a and b
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

use long division
......(x%5E2%2B%28a+%2B1%29x%2B%28b%2Ba+%2B1%29
x-1| x%5E3%2Bax%5E2%2Bbx-3
........x%5E3-x%5E2
..............ax%5E2%2Bx%5E2
..............%28a+%2B1%29x%5E2%2Bbx
..............%28a+%2B1%29x%5E2-%28a+%2B1%29x+
..............................bx%2B%28a+%2B1%29x+-3
................................%28b%2Ba+%2B1%29x+-%28b%2Ba+%2B1%29
........................................... -3%2Bb%2Ba+%2B1
........................................... +b%2Ba+-2+=> reminder+b%2Ba+-2+=1=>b%2Ba+=3..........eq.1


......(x%5E2%2B%281+-a%29x%2B%28b%2B1+-a%29
x%2B1| x%5E3%2Bax%5E2%2Bbx-3
........x%5E3%2Bx%5E2
..............ax%5E2-x%5E2
...............%281+-a%29x%5E2%2Bbx
...............%281+-a%29x%5E2-%281+-a%29x+-3
..........................bx%2B%281+-a%29x+%2B%28b%2B1+-a%29
................................................-3-%28b%2B1+-a%29+
................................................. -3-b-1%2Ba++
................................................ +a-b+-4 => reminder +a-b+-4+=-9=> a-b+=-5..........eq.2

go to
b%2Ba+=3..........eq.1, solve for a
a=3-b
go to
a-b+=-5..........eq.2, substitute a
3-b-b+=-5
3-2b+=-5
3%2B5+=2b
2b=8
b=4
then
a=3-b => a=3-4 => a=-1

your equation is
f%28x%29=x%5E3-1x%5E2%2B4x-3

check reminders:
x%5E3+-+x%5E2+%2B+4+x+-+3+=+%28x%5E2+%2B+4%29%2A%28x+-+1%29+%2B+1
x%5E3+-+x%5E2+%2B+4+x+-+3+=+%28x%5E2+-+2+x+%2B+6%29%2A%28x+%2B+1%29+%2B+%28-9%29



Answer by ikleyn(52798) About Me  (Show Source):
You can put this solution on YOUR website!
.
Given that f(x) = x^3 + ax^2 + bx - 3 leaves remainders 1 and -9 when divided by x-1 and x+1.
Calculate the values of a and b
~~~~~~~~~~~~~~~~~~ 


Since  f(x) = x^3 + ax^2 + bx - 3 leaves remainder 1 when divided by x-1, it means,

according to the remainder theorem, that  f(1) = 1,   or


    1^3 + a*1^2 + b*1 - 3 = 1,

or

    a + b = 1 - 1 + 3,

    a + b = 3.      (1)



Since  f(x) = x^3 + ax^2 + bx - 3 leaves remainder -9 when divided by x+1, it means,

according to the remainder theorem, that  f(-1) = -9,   or


    (-1)^3 + a*(-1)^2 + b*(-1) - 3 = -9,

or

    a - b = 1 + 3 -9,

    a - b = -5.     (2)



Thus we have two equations (1) and (2) to find two unknowns "a" and "b".

So we add these equations, and we get  


    2a = 3 + (-5) = -2,  hence,  a = (-2)/2 = -1.


Then from equation (1),  b = 3 - a = 3 - (-1) = 4.


ANSWER.  a= -1;  b= 4.


Solved.