Determining values of Coefficients

Question

I have tried to figure this question out but I am not sure if i'm doing it correctly. If somebody could help explain it would be appreciated.

Question

Determine the values of m and n for $f(x)= mx^3+20x^2+nx-35$ given that (x+1) gives a remainder of zero, and when divided by (x-2) the remainder is 45.

Not sure if this is correct.

$f(-1)=m(-1)^3+20(-1)^2+n(-1)-35$

$=-1m-1n+20-35$

$=-1m-1n-15$

ok so then what should i do?

$f(x)=m(2)^3+20(2)+n(2)-35$

$=8m+2n+80-35$

$=8m+2n+45$

If you could can you explain your answer? I have been trying to figure out how you came to your answer but I don't understand. thankyou

Answer 1

$f(x)=(x+1)g(x)$ implies that $f(-1)=0$, and you've shown this implies $-m-n-15=0$.

$f(x)=(x-2)h(x)+45$ implies $f(2)=45$, and you've shown this implies $8m+2n + 45=45$.

The second equation implies $n=-4m$ and plugging this into the first yields $m=-5$ and $n=20$.