Find the constant c such that the denominator divides evenly into the numerator.
(x^4 - 3x^2 + c)/( x+6)
I did a synthetic division (or you can do a long algebraic division) to get a remainder of (c+1188)
So for the division to be exact,
x+1188 = 0
c = - 1188
http://www.wolframalpha.com/input/?i=%28x%5E4+-+3x%5E2+-1188%29%2F%28+x%2B6%29
look at "quotient and remainder" result
To find the constant c such that the denominator x+6 divides evenly into the numerator x^4 - 3x^2 + c, you can use polynomial long division. Here's how you can do it:
1. Set up the long division by dividing the first term of the numerator (x^4) by the denominator (x+6). Write the quotient on top and the divisor on the left side of a long division bracket, like this:
_______
x + 6 | x^4 - 3x^2 + c
2. Divide the first term of the numerator (x^4) by the denominator (x), which gives you x^3. Write the result (x^3) on top of the division bracket, above the line.
x^3
_______
x + 6 | x^4 - 3x^2 + c
3. Multiply the divisor (x + 6) by the quotient (x^3) and write the result (x^3 * (x + 6) = x^4 + 6x^3) below the line, aligned with the terms of the numerator.
x^3
_______
x + 6 | x^4 - 3x^2 + c
- (x^4 + 6x^3)
4. Subtract the result (x^4 + 6x^3) from the numerator (x^4 - 3x^2 + c) and write the difference (-3x^2 - 6x^3) below the line.
x^3
_______
x + 6 | x^4 - 3x^2 + c
- (x^4 + 6x^3)
__________
- 6x^3 - 3x^2 + c
5. Bring down the next term of the numerator (-3x^2) and continue the long division process.
x^3 - 6x^2
_______
x + 6 | x^4 - 3x^2 + c
- (x^4 + 6x^3)
__________
- 6x^3 - 3x^2 + c
6. Divide the new term (-6x^2) by the denominator (x), which gives you -6x. Write the result (-6x) above the line.
x^3 - 6x - 6
_______
x + 6 | x^4 - 3x^2 + c
- (x^4 + 6x^3)
__________
- 6x^3 - 3x^2 + c
7. Multiply the divisor (x + 6) by the new quotient (-6x) and write the result (-6x * (x + 6) = -6x^2 - 36x) below the line, aligned with the terms of the numerator.
x^3 - 6x - 6
_______
x + 6 | x^4 - 3x^2 + c
- (x^4 + 6x^3)
__________
- 6x^3 - 3x^2 + c
+ ( -6x^2 - 36x)
8. Subtract the result (-6x^2 - 36x) from the current numerator (-6x^3 - 3x^2 + c) and write the difference (33x^2 - 36x + c) below the line.
x^3 - 6x - 6
_______
x + 6 | x^4 - 3x^2 + c
- (x^4 + 6x^3)
__________
- 6x^3 - 3x^2 + c
+ (-6x^2 - 36x)
_____________
33x^2 - 36x + c
9. Lastly, check if the new numerator (33x^2 - 36x + c) is divisible by the denominator (x + 6). If it is, the constant c would be such that the denominator divides evenly into the numerator.
So, to determine the value of the constant c, you need to equate the new numerator to zero (since the denominator divides evenly into the numerator). Therefore, set 33x^2 - 36x + c = 0 and solve for c using quadratic equation methods or factoring techniques.