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ratelena [41]
3 years ago
15

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

Mathematics
1 answer:
Alla [95]3 years ago
7 0

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.

\dfrac{3m+3n}{3}=\dfrac{-18}{3}

m+n=-6

Hence proved.

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