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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

Verified Answer :

LammettHash

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,

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

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