Given:
The expression is:
[tex]2x^3+mx^2+nx+c[/tex]
It leaves the same remainder when divided by x -2 or by x+1.
To prove:
[tex]m+n=-6[/tex]
Solution:
Remainder theorem: If a polynomial P(x) is divided by (x-c), thent he remainder is P(c).
Let the given polynomial is:
[tex]P(x)=2x^3+mx^2+nx+c[/tex]
It leaves the same remainder when divided by x -2 or by x+1. By using remainder theorem, we can say that
[tex]P(2)=P(-1)[/tex] ...(i)
Substituting [tex]x=-1[/tex] in the given polynomial.
[tex]P(-1)=2(-1)^3+m(-1)^2+n(-1)+c[/tex]
[tex]P(-1)=-2+m-n+c[/tex]
Substituting [tex]x=2[/tex] in the given polynomial.
[tex]P(2)=2(2)^3+m(2)^2+n(2)+c[/tex]
[tex]P(2)=2(8)+m(4)+2n+c[/tex]
[tex]P(2)=16+4m+2n+c[/tex]
Now, substitute the values of P(2) and P(-1) in (i), we get
[tex]16+4m+2n+c=-2+m-n+c[/tex]
[tex]16+4m+2n+c+2-m+n-c=0[/tex]
[tex]18+3m+3n=0[/tex]
[tex]3m+3n=-18[/tex]
Divide both sides by 3.
[tex]\dfrac{3m+3n}{3}=\dfrac{-18}{3}[/tex]
[tex]m+n=-6[/tex]
Hence proved.
