Question

The polynomial `6x^3+mx^2+nx-5` has a factor of x+1. when divided by x=1, the remainder is -4. What are the values of m and n?

The polynomial `6x^3+mx^2+nx-5` has a factor of x+1. when divided by x=1, the remainder is -4. What are the values of m and n?

Answer 1

`m=7` and `n=-4`

Explanation:

I'm going to assume that when you write "when divided by `x=1`" you mean when dividing by the monomial `x - 1`.

If `x-1` is a factor that means that when `x=-1`, the function goes to zero. Hence,
`6 (-1)^3 + m (-1)^2 + n (-1) - 5 = 0 implies m - n = 11`

If we divide by `x-1` and have a remainder of 4, that means that the value of the expression at `x=1` is 4. Therefore,
`6 (1)^3 + m(1)^2 + n (1) - 5 = 4 implies m + n = 3`

We have two equations and two unknowns, hence we can solve for `m` and `n`. We add then and divide by two to find that `m = 7` and `n = -4`.

SIDENOTE:
If we want to prove the second piece of logic (that `x=1` makes the expression 4), we can imagine the expression minus 4 to now have a factor of `x-1` (by definition of remainder) and we can use the logic from above. Hope that makes sense!