Question

if (x+1) and (x-2) are factors of x^4-x^3+mx^2+nx+4 find the values of m and n​

Answer 1

Answer:

well for sure, you need some extra data because with just 2 factors, you cannot reach x^3, and, because of that you may have, if possible any number of them. but lets not do it so fast.

(x+4).(x-1). f(x) = x^3 + m.x^2 -n.x + 8 -> solving the left side.

x^2+3.x-4. f(x) = x^3 + m.x^2 -n.x + 8

with that we can notice that at least (if not the only) way to solve this is that f(x) = (x - 2)

why this because if f(x) = (x-2)

the left side is now.

x^3 + 3.x^2–4.x -2.x^2 -6.x +8 = x^3 + m.x^2 -n.x + 8

from this point we can simplify x^3 and 8 in both sides. and add all that has the same order of x.

+ 3.x^2–4.x -2.x^2 -6.x = + m.x^2 -n.x -> x^2 - 10 x = m.x^2 - n.x

this is the same as saying, x^2=m.x^2 and -10.x = -n.x

and there you simple have m=1 and n=10.