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 ??

1 Answer
Douglas K.
Oct 29, 2016

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#

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