Yuzrah's class was given an assignment to solve for the zeroes of a 5th-degree monic polynomial function. Her attention-seeker partner turned off the projector and thus, she was able to write only P(x) = x⁵ + 20x⁴ + 20x³, forgetting the x², x, and constant terms. However, her professor spoiled that the roots formed an arithmetic progression.
Q1. What are the roots of P(x)?
Q2. What is P(x)?
We know the roots are m, m+d, and m+2d, ... an arithmetic progression. So, we have
P(x) = (x-m)(x-(m+d))(x-(m+2d))(x-(m+3d))(x-(m+4d))
= x^5 + (-5m-10d)x^4 + (10m^2+40md+35d^2)x^3 + ...
So now we know that
-5m-10d = 20
10m^2+40md+35d^2 = 20
Unfortunately, that yields an irrational solution, as
m = 4(√7-1) ≈ 6.5830
d = -2√7 ≈ -5.2915
Plugging that in, we get
P(x) = x^5 + 20x^4 + 20x^3 - 1040x^2 - 2304x + 4608
I suspect a typo somewhere; that's just too much work!