Question 935800

Find the function {{{f(x)=ax^3+bx^2+cx+d}}} for which {{{f(0)=-2}}}, {{{f(1)=5}}}, {{{f(-1)=3}}}, {{{f(2)=4}}}:

{{{f(x)=ax^3+bx^2+cx+d}}}....we need to find coefficients

{{{f(0)=-2}}}

{{{f(0)=a*0^3+b*0^2+c*0-2}}}=> {{{highlight(d=-2)}}}


{{{f(1)=5}}}

{{{5=a*1^3+b*1^2+c*1-2}}}

{{{5+2=a+b+c}}}

{{{a+b+c=7}}}..................eq.1


{{{f(-1)=3}}}

{{{3=a*(-1)^3+b*(-1)^2+c*(-1)-2}}}

{{{3+2=-a+b-c}}}

{{{-a+b-c=5}}}..................eq.2


{{{f(2)=4}}}

{{{4=a*2^3+b*2^2+c*2-2}}}

{{{4+2=8a+4b+2c}}}

{{{6=8a+4b+2c}}}

{{{3=4a+2b+c}}}

{{{4a+2b+c=3}}}..................eq.3


use eq.1 and 2

{{{a+b+c=7}}}..................eq.1
{{{-a+b-c=5}}}..................eq.2
------------------------------------add both

{{{a+b+c-a+b-c=7+5}}}
{{{2b=12}}}
{{{highlight(b=6)}}}

go to {{{4a+2b+c=3}}}..................eq.3 substitute {{{b}}} 

{{{4a+2*6+c=3}}}

{{{4a+12+c=3}}}...solve for {{{c}}}

{{{c=3-4a-12}}}

{{{c=-4a-9}}}....->

go to {{{a+b+c=7}}}..................eq.1 substitute {{{b}}} and {{{c}}}

{{{a+6-4a-9=7}}}.....solve for {{{a}}}

{{{-3a-3=7}}}

{{{-3a=7+3}}}

{{{highlight(a=-10/3)}}}

then {{{c=-4a-9}}}

{{{c=-4(-10/3)-9}}}

{{{c=40/3-27/3}}}

{{{highlight(c=13/3)}}}


the function is: {{{f(x)=-(10/3)x^3+6x^2+(13/3)x-2}}}