Question

Help please : Polonymal :

Question : Find a,b and c.

2x^3 +3x^2 -4x -5 = a(x+1)^3 +b(x+1)^2 +c(x+1)

Thanks for explaining

Answer 1

Expand the right hand side, simplify, and match up the coefficients of same-power terms.


`a(x+1)^3+b(x+1)^2+c(x+1)=a(x^3+3x^2+3x+1)+b(x^2+2x+1)+c(x+1)`

`=ax^3+(3a+b)x^2+(3a+2b+c)x+(a+b+c)`

Since this must be the same as
`2x^3+3x^2-4x-5`, you need to satisfy


`\begin{cases}a=2\\3a+b=3\\3a+2b+c=-4\\a+b+c=-5\end{cases}`

The solution for
`a` is already given,
`a=2`. Substitute this into the second equation to find
`b`.


`3a+b=3\implies 6+b=3\implies b=-3`

Now plug
`a,b` into either remaining equation to find
`c`.


`a+b+c=-5\implies 2-3+c=-5\implies c=-4`

And just to check that the third equation is also satisfied,


`3a+2b+c=-4\implies 6-6-4=-4\checkmark`