Question

Urgent! Help! A cubic polynomial has zeros at x=-1, x=1, and x=3. It has a y-intercept of -6. What is the remainder when we divide this polynomial by x^2+1 ??

Answer 1

I used WolframAlpha to do the division the remainder is `4x - 12`

Explanation:

Start is a factor, k that allows one to adjust the y intercept:

k

Multiply that by the factor corresponding to the zero, `x = -1` -- that is `(x + 1)`

`k(x + 1)`

Multiply by the factor corresponding to the zero, `x = -1` -- that is `(x - 1)`

`k(x + 1)(x - 1)`

The last factor is the one corresponding to the zero, `x = 3` -- `(x - 3)`

`k(x + 1)(x - 1)(x - 3)`

To find the value of k, we set the factors equal to -6 and x within the factors equal to 0

`-6 = k(+1)(-1)(-3)`

`k = -2`

Because the divisor is not of the form `(x - a)` but, instead, of the form `(x^2 - a)`, one cannot use the remainder theorem. Therefore, the only way to find the remainder is by using division.

Here is a link to WolframAlpha for the division.

Please notice that the remainder is `4x - 12`