Question 1155341
Find the function ​f(x)=ax^3+bx^2+cx+d for which, f(-3)=-83, f(-1)=1, f(1)=5,
f(2)=7.
<pre>

{{{system(matrix(5,5,
ax^3+bx^2+cx+d,""="","f(x)","","",
a(-3)^3+b(-3)^2+c(-3)+d,""="","f(-3)",""="",-83,
a(-1)^3+b(-1)^2+c(-1)+d,""="","f(-1)",""="",1,
a(1)^3+b(1)^2+c(1)+d,""="","f(1)",""="",5,
a(2)^3+b(2)^2+c(2)+d,""="","f(2)",""="",7  ))}}}

That simplifies to this system of equations:

{{{system(
matrix(4,9,
-27a,""+"",9b,""-"",3c,""+"",d,""="",-83,
-a,""+"",b,""-"",c,""+"",d,""="",1,
a,""+"",b,""+"",c,""+"",d,""="",5,
8a,""+"",4b,""+"",2c,""+"",d,""="",7 ))}}}

Solve that system either by elimination or 
matrix methods.

a=2, b=-4, c=0, d=7

So:

{{{matrix(1,3,"f(x)",""="",ax^3+bx^2+cx+d)}}}

becomes:

{{{matrix(1,3,"f(x)",""="",2x^3-4x^2+0x+7)}}}

We drop the 0x term:

{{{matrix(1,3,"f(x)",""="",2x^3-4x^2+7)}}}

Edwin</pre>