SOLUTION: Write a polynomial division that has a quotient of x+5 and a remainder -2

Question 1198625: Write a polynomial division that has a quotient of x+5 and a remainder -2

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

n = numerator
d = denominator
q = quotient = x+5
r = -2 = remainder

n/d = q + (r/d)
n = d*q + r
n = d*(x+5) + (-2)
n = d*(x+5) - 2

Now pick anything you like for the denominator d.
I find its easiest to select a binomial.
I'll go for d = x+1.

n = d*(x+5) - 2
n = (x+1)*(x+5) - 2
n = (x^2+5x+1x+5) - 2
n = x^2+6x+3

Therefore, dividing (x^2+6x+3)/(x+1) will result in a quotient of x+5 and remainder -2.
I'll let you use either polynomial long division or synthetic division to confirm this claim.

Infinitely many rational functions of the form n/d will lead to q = x+5 and r = -2.
In other words, (x^2+6x+3)/(x+1) isn't the only possibility here.

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