Hello Steven,
I add a few words to the previous expert's answer.
Rational Roots Theorem. If p/q is a rational root in lowest terms of the polynomial equation with integer coefficients anxn + ... + a1x + a0 = 0 (where an ≠ 0 and a0 ≠ 0), then p divides a0 and q divides an.
So, if p/q is a rational zero of f(x) = 7x4 - 5x3 + x2 - x + 1, then p divides 1 and q divides 7.
p is an integer, q is a natural number, so p could be only ±1, q could be only 1 or 7.
Choices A and B are excluded because they have denominator 5.
Choice C is excluded because it has numerator 7.
Only choice D remains.
Comment. Actually, all choices are wrong because ±1 are not rational roots (and ±1/7 are not rational roots also).
Vitaliy V.
12/10/21
Dayv O.
we agree one is a rational number. In a problem like this for student it is important they understand the size of the set of numbers that can be roots. Kind of just like you say that is how many times a test need to be computed to see if value solves f(x)=0 equation or has no remainder in synthetic division . They should understand set size does not depend on order of polynomial. They should understand the rule for each potential member. Concluding with proper set size with the proper potential solutions as members of set. Then and only then can they proceed to test as you say for viability of correctness. The problem as we agree just wants the set of possible solutions. D. we agree is correct answer ---and I agree "±1 are not rational roots (and ±1/7 are not rational roots also)".. for the given function.12/10/21
Dayv O.
could you explain how 1 and 1/7 are not rational. The problem only asks for potential roots.12/09/21