Question 1077691
{{{f(x) = ax^4 + bx^3 + cx^2 + dx + e }}}

{{{ f(0) = 1 = 0+0+0+0+e }}}   —> {{{ e = 1 }}}  (use e=1 in evaluating f(1)…f(4) below)
{{{ f(1) = 1 }}} —> {{{ a + b + c + d + e = 1 }}} —> {{{a + b + c + d = 0 }}}
{{{ f(2) = 1 }}} —> {{{ a(16) + b(8) + c(4) + d(2) + e = 1 }}} —> {{{ 16a + 8b + 4c + 2d = 0 }}}
{{{ f(3) = 1 }}} —> ... —> {{{ 81a + 27b + 9c + d = 0 }}}
{{{ f(4) = 0 }}} —> … —> {{{ 256a + 64b + 16c + 4d + 1 = 0 }}} 
—
This generates the following matrix (moving constants to right hand side):
—
{{{ (matrix(4,4, 1,1,1,1, 16,8,4,2,  81,27,9,3,  256,64,16,4)) * (matrix(4,1, a,b,c,d)) = (matrix(4,1, 0,0,0,-1)) }}}

The inverse of the 4x4 matrix is (from https://www.symbolab.com/solver/matrix-inverse-calculator )

{{{ (matrix(4,4,  -1/6, 1/4, -1/6, 1/24,   3/2,-2, 7/6, -1/4,  -13/3, 19/4, -7/3, 11/24,   4,-3,4/3,-1/4))  }}} 
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And the solution we seek is:
{{{ (matrix(4,1,a,b,c,d)) = (matrix(4,4,  -1/6, 1/4, -1/6, 1/24,   3/2,-2, 7/6, -1/4,  -13/3, 19/4, -7/3, 11/24,   4,-3,4/3,-1/4))*(matrix(4,1, 0,0,0,-1)) }}} 




{{{ a = -1/6 + 1/4 - 1/6 - 1/24 = -3/24 }}}
{{{ b = 3/2 - 2 + 7/6 + 1/4 = 22/24 }}}
{{{ c = -57/24 }}}
{{{ d = 62/24 }}}

{{{f(x) = (-3/24)x^4 + (22/24)x^3 + (-57/24)x^2 + (62/24)x + 1 }}}
pull out 1/24:
{{{f(x) = (1/24)(-3x^4 + 22x^3 -57x^2 + 62x + 24) }}}
—
Now we are ready to find f(5):
{{{ f(5) = (1/24)(-3(625) + 22(125) -57(25) + 62(5) + 24) }}}
{{{ f(5) = (1/24)(-216) = highlight(-9) }}}