When f(x)=2x^3+ax^2+bx+c is divided by (x^2-1) the remainder is x+1. If (x-2) is a factor of f(x), find the values of the constants a,b and c.

Thanks :)

since x^2 - 1 = (x+1)(x-1)

for both f(1) and f(-1) we get a remainder of x+1

f(1) = 2+a+b+c = 1+1
a+b+c = 0 (#1)

f(-1) = -2+a-b+c = 0
a-b+c = 2 (#2)

also x-2 is a factor, so f(2) = 0
f(2) = 16+4a+2b+c = 0
4a + 2b+c = -16 (#3)
#1 - #2 ---> 2b=-2 , b = -1

new #3 : 4a -2 + c = -16 or 4a + c = -14

new #1 : a - 1 + c = 0 or a + c = 1

new #3 - new #1
3a = -15
a = -5
in #1
-5 + c = 1
c = 6

original #1
a+b+c=0
a-1+6=0
a=-5

a= -5
b= -1
c= 6

thanks :)

To find the values of the constants a, b, and c, we can use the concept of polynomial division.

Step 1: Perform polynomial division
Divide f(x) by (x^2-1) and set the remainder equal to x+1.
So, the polynomial division can be represented as:
2x^3 + ax^2 + bx + c = (x^2 - 1)(q(x)) + (x+1)
where q(x) is the quotient when f(x) is divided by (x^2-1).

Step 2: Solve the quotient
Now, let's solve for q(x) by dividing 2x^3 + ax^2 + bx + c by x^2 - 1.

____________________________
x^2 - 1 | 2x^3 + ax^2 + bx + c

To get the quotient, we divide the leading terms of the numerator and denominator, which are 2x^3 and x^2 respectively. So, the quotient term is 2x.

2x
____________________________
x^2 - 1 | 2x^3 + ax^2 + bx + c

Next, we multiply the divisor (x^2 - 1) by the quotient term (2x) and subtract it from the numerator:
(2x) * (x^2 - 1) = 2x^3 - 2x
Subtracting this from the numerator, we get:
(2x^3 + ax^2 + bx + c) - (2x^3 - 2x) = ax^2 + bx + c + 2x

Now, we repeat the process by dividing the new numerator (ax^2 + bx + c + 2x) by the divisor (x^2 - 1).

_____________________________
x^2 - 1 | ax^2 + bx + c + 2x

Following the same steps as before, we find that the quotient term is a.

2x + a
_____________________________
x^2 - 1 | ax^2 + bx + c + 2x

Multiplying the divisor (x^2 - 1) by the quotient term (a), we have:
(a) * (x^2 - 1) = ax^2 - a
Subtracting this from the numerator, we get:
(ax^2 + bx + c + 2x) - (ax^2 - a) = bx + c + 2x + a

Repeat the process by dividing the new numerator (bx + c + 2x + a) by the divisor (x^2 - 1).

__________________________________
x^2 - 1 | bx + c + 2x + a

The quotient term is b.
2x + a + b
__________________________________
x^2 - 1 | bx + c + 2x + a

Multiplying the divisor (x^2 - 1) by the quotient term (b), we have:
(b) * (x^2 - 1) = bx^2 - b
Subtracting this from the numerator, we get:
(bx + c + 2x + a) - (bx^2 - b) = c + 2x + a - bx^2 + b

Since there's no polynomial left in the numerator, the division is complete.
So, the remainder should be equal to x + 1:
c + 2x + a - bx^2 + b = x + 1

Step 3: Find the values of a, b, and c.
Now, equate the coefficients of like terms on both sides of the equation.

From the x^2 term:
- b = 0 --> b = 0

From the x term:
2 = 1 --> a = 1

From the constant term:
c + a + b = 1 --> c + 1 + 0 = 1 --> c + 1 = 1 --> c = 0

Therefore, the values of the constants are:
a = 1, b = 0, c = 0.

So, the values of the constants a, b, and c are 1, 0, and 0 respectively.