Question

Given that the expression 2x^3 + mx^2 + nx + c leaves the same remainder when divided by x -2 or by x+1 I prove that m+n =-6

Answer 1

Given:

The expression is:

`2x^3+mx^2+nx+c`

It leaves the same remainder when divided by x -2 or by x+1.

To prove:

`m+n=-6`

Solution:

Remainder theorem: If a polynomial P(x) is divided by (x-c), thent he remainder is P(c).

Let the given polynomial is:

`P(x)=2x^3+mx^2+nx+c`

It leaves the same remainder when divided by x -2 or by x+1. By using remainder theorem, we can say that

`P(2)=P(-1)` ...(i)

Substituting
`x=-1` in the given polynomial.

`P(-1)=2(-1)^3+m(-1)^2+n(-1)+c`

`P(-1)=-2+m-n+c`

Substituting
`x=2` in the given polynomial.

`P(2)=2(2)^3+m(2)^2+n(2)+c`

`P(2)=2(8)+m(4)+2n+c`

`P(2)=16+4m+2n+c`

Now, substitute the values of P(2) and P(-1) in (i), we get

`16+4m+2n+c=-2+m-n+c`

`16+4m+2n+c+2-m+n-c=0`

`18+3m+3n=0`

`3m+3n=-18`

Divide both sides by 3.

`(3m+3n)/(3)=(-18)/(3)`

`m+n=-6`

Hence proved.