SOLUTION: 1.)When the expression x^3+mx^2+nx-5 is divided by x+2 and x+3, the remainders are 2 and 13 respectively. Find the values of m and n. 2)When P(x)=x^2+ax-b is divided by x-1, the

Algebra ->  Expressions -> SOLUTION: 1.)When the expression x^3+mx^2+nx-5 is divided by x+2 and x+3, the remainders are 2 and 13 respectively. Find the values of m and n. 2)When P(x)=x^2+ax-b is divided by x-1, the      Log On


   



Question 988995: 1.)When the expression x^3+mx^2+nx-5 is divided by x+2 and x+3, the remainders are 2 and 13 respectively. Find the values of m and n.
2)When P(x)=x^2+ax-b is divided by x-1, the remainder is 6. When P(x) is divided by x+2, the remainder is 3. Find the values of a and b.
Please help me and thank you

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
When a polynomial P%28x%29 is divided by x-k (where k is any real number) is P%28k%29 .

1) P%28-2%29=%28-2%29%5E3%2Bm%28-2%29%5E2%2Bn%28-2%29-5=-8%2B4m-2n-5=4m-2n-13
If the remainder is 2 when P%28x%29=x%5E3%2Bmx%5E2%2Bnx-5 is divided by x%2B2=x-%28-2%29%29 ,
then 4m-2n-13=2--->4m-2n=15 .
P%28-3%29=%28-3%29%5E3%2Bm%28-3%29%5E2%2Bn%28-3%29-5=-27%2B9m-3n-5=9m-3n-32
If the remainder is 13 when P%28x%29=x%5E3%2Bmx%5E2%2Bnx-5 is divided by x%2B3=x-%28-3%29%29 ,
then 9m-3n-32=13--->9m-3n=45--->3m-n=15--->n=3m-15 .
system%28n=3m-15=45%2C4m-2n=15%29--->system%28n=3m-15=45%2C4m-2%283m-15%29=15%29--->system%28n=3m-15=45%2C4m-6m%2B30=15%29
--->system%28n=3m-15=45%2C-2m%2B30=15%29--->system%28n=3m-15=45%2C-2m-30=15%29--->system%28n=3m-15=45%2Cm=15%2F2%29
--->system%28n=3%2815%2F2%29-15=45%2Cm=15%2F2%29--->system%28n=45%2F2-30%2F2=45%2Cm=15%2F2%29--->highlight%28system%28n=15%2F2%2Cm=15%2F2%29%29

2) Similarly,
P%281%29=1%5E2%2Ba%2A1-b=1%2Ba-b=6--->a-b=6-1--->a-b=5
and P%28-2%29=%28-2%29%5E2%2Ba%2A%28-2%29-b=4-2a-b=3--->-2a-b=3-4--->-2a-b=-1--->2a%2Bb=1
system%28a-b=5%2C2a%2Bb=1%29--->system%28a-b=5%2Ca-b%2B2a%2Bb=5%2B1%29--->system%28a-b=5%2C3a=6%29--->system%28a-b=5%2Ca=2%29--->system%282-b=5%2Ca=2%29--->system%28-b=3%2Ca=2%29--->highlight%28system%28b=-3%2Ca=2%29%29